The document discusses parametric equations of lines. It introduces scalar multiplication and the base to tip rule for vectors. It then shows that a line L passing through a point P in the direction of a vector D can be represented parametrically as L = P + tD, where t is a real number. Specifically, if P = (c,d) and D = (a,b), then the parametric equations for L are x = c + at, y = d + bt. An example is given of finding parametric equations for a line with given direction and point.
Vectors can be represented as arrows and describe direction and magnitude. They are used to represent physical quantities like forces and movements. A vector is defined by its base point and tip or by its coordinates in a system. Vector addition and subtraction follow graphical rules like head-to-tail, with the sum of vectors forming a parallelogram. Vectors can be multiplied by scalars to change their length and direction.
The document discusses the dot product of vectors and its relationship to the angle between vectors. It states that for two unit vectors u and v, the cosine of the angle between them is equal to their dot product (u - v). The dot product is defined as the sum of the products of corresponding components (ac + bd). Several examples are also given to demonstrate calculating the dot product and how it relates to the angle between vectors.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses arc length parameterization and curvature of curves. It gives an example of a curve C(t) = <3t + 1, 4t - 2> describing a particle moving at a constant speed of 5. Reparameterizing this curve by arc length gives C*(s) = <3/5s + 1, 4/5s - 2>, which describes the particle moving at a speed of 1. The relationship between s and t is determined using the arc length formula. The curvature of a curve measures its rate of turning, with straight lines having curvature of 0 and circles having curvature of 1/r, where r is the radius.
The document defines unit tangent and normal vectors for vector-valued functions. The unit tangent vector T(t) is the normalized derivative of the vector function C(t). The principal unit normal vector N is defined as the derivative of T(t) normalized. In 2D, there are two normal vectors to the tangent vector C'(t); the document provides formulas to determine them. Examples are given to demonstrate calculating T(t) and N for a given vector function at a specified time.
The document discusses graph algorithms and graph theory concepts. It defines graphs, directed and undirected graphs, and graph terminology like vertices, edges, adjacency, paths, cycles, connectedness, and representations using adjacency lists and matrices. It also covers tree graphs and their properties.
This document provides an overview of basic graph algorithms. It begins with examples of graphs in everyday life and a brief history of graph theory starting with Euler. It then covers basic graph terminology and properties like nodes, edges, degrees. Common representations of graphs in computers like adjacency lists and matrices are described. Breadth-first search and depth-first search algorithms for traversing graphs are introduced. Finally, applications of graph algorithms like finding paths, connected components, and topological sorting are mentioned.
This document discusses polyadic geometry, which uses polyadic calculus to represent physical laws through tensors and free polyadics of different valences. It explains that polyadics of valence zero are scalars, valence one are vectors, and higher valences (H) are H-adics like dyadics and triadics. The document outlines how polyadic spaces are conceived similarly to vector spaces, with dimensions of an H-adic space being 3H. It also describes how to construct geometric figures like pyramids in polyadic spaces and extend concepts like trigonometry and medians to higher dimensions using polyadic representations.
Vectors can be represented as arrows and describe direction and magnitude. They are used to represent physical quantities like forces and movements. A vector is defined by its base point and tip or by its coordinates in a system. Vector addition and subtraction follow graphical rules like head-to-tail, with the sum of vectors forming a parallelogram. Vectors can be multiplied by scalars to change their length and direction.
The document discusses the dot product of vectors and its relationship to the angle between vectors. It states that for two unit vectors u and v, the cosine of the angle between them is equal to their dot product (u - v). The dot product is defined as the sum of the products of corresponding components (ac + bd). Several examples are also given to demonstrate calculating the dot product and how it relates to the angle between vectors.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
The document discusses arc length parameterization and curvature of curves. It gives an example of a curve C(t) = <3t + 1, 4t - 2> describing a particle moving at a constant speed of 5. Reparameterizing this curve by arc length gives C*(s) = <3/5s + 1, 4/5s - 2>, which describes the particle moving at a speed of 1. The relationship between s and t is determined using the arc length formula. The curvature of a curve measures its rate of turning, with straight lines having curvature of 0 and circles having curvature of 1/r, where r is the radius.
The document defines unit tangent and normal vectors for vector-valued functions. The unit tangent vector T(t) is the normalized derivative of the vector function C(t). The principal unit normal vector N is defined as the derivative of T(t) normalized. In 2D, there are two normal vectors to the tangent vector C'(t); the document provides formulas to determine them. Examples are given to demonstrate calculating T(t) and N for a given vector function at a specified time.
The document discusses graph algorithms and graph theory concepts. It defines graphs, directed and undirected graphs, and graph terminology like vertices, edges, adjacency, paths, cycles, connectedness, and representations using adjacency lists and matrices. It also covers tree graphs and their properties.
This document provides an overview of basic graph algorithms. It begins with examples of graphs in everyday life and a brief history of graph theory starting with Euler. It then covers basic graph terminology and properties like nodes, edges, degrees. Common representations of graphs in computers like adjacency lists and matrices are described. Breadth-first search and depth-first search algorithms for traversing graphs are introduced. Finally, applications of graph algorithms like finding paths, connected components, and topological sorting are mentioned.
This document discusses polyadic geometry, which uses polyadic calculus to represent physical laws through tensors and free polyadics of different valences. It explains that polyadics of valence zero are scalars, valence one are vectors, and higher valences (H) are H-adics like dyadics and triadics. The document outlines how polyadic spaces are conceived similarly to vector spaces, with dimensions of an H-adic space being 3H. It also describes how to construct geometric figures like pyramids in polyadic spaces and extend concepts like trigonometry and medians to higher dimensions using polyadic representations.
The document defines sequences and series, and describes three types of progressions: arithmetic, geometric, and harmonic. It then focuses on arithmetic progressions (AP), defining them as sequences where the difference between consecutive terms is constant. The nth term of an AP is given as an + (n-1)d, where a is the first term and d is the common difference. The sum of the first n terms of an AP is provided as (2a + (n-1)d)/2. Arithmetic means are also discussed, along with finding the number of arithmetic means between two numbers in an AP.
The document defines sequences and series, and describes three types of progressions: arithmetic, geometric, and harmonic. It then focuses on arithmetic progressions (AP), defining them as sequences where the difference between consecutive terms is constant. The nth term of an AP is given as an + (n-1)d, where a is the first term and d is the common difference. The sum of the first n terms of an AP is provided as (2a + (n-1)d)/2. Arithmetic means are also discussed, along with finding the number of arithmetic means between two numbers in an AP.
Vectors can be represented as arrows with a magnitude (length) and direction. They are used in mathematics and physics to represent numerical measurements in a specified direction. There are rules for vector addition and scalar multiplication that involve combining or extending/compressing vectors. Vectors can be added using the parallelogram rule or the base-to-tip rule, with the sum being independent of order for three or more vectors.
This document provides information about sequences and series in mathematics. It defines a sequence as a function whose domain is the set of natural numbers and whose range is a set of term values. Examples of finding the next term in sequences are provided. Summation notation is introduced to write the terms of a series and evaluate its sum. Convergent and divergent sequences are defined. Properties of sequences and summation are outlined, including using Desmos to list terms of a sequence. Examples are provided to demonstrate evaluating finite and infinite series using summation notation and properties. Exercises for students are listed at the end.
Squares have the same meaning as quadratic and refer to something that is multiplied by itself or involves terms of the second degree. When you square a number, it is multiplied by itself to give a positive result. A square is also a specific geometric shape with four equal sides that meet at right angles. The area of a square can be calculated by multiplying the length of its sides or taking half of the diagonal squared, while the perimeter is four times the length of its sides.
Vectors are numerical measurements of direction and magnitude. They are represented geometrically by arrows, with the length indicating magnitude and direction pointing the tip. Vector addition follows the parallelogram rule, where the resultant vector is the diagonal of the parallelogram formed by the two vectors. It can also be done by the base to tip rule, placing the base of one vector at the tip of the other. Scalar multiplication scales the vector by a numerical factor, extending or compressing its length.
The Floyd-Warshall algorithm finds the shortest paths between all pairs of vertices in a weighted graph. It does this by considering all possible paths through intermediate vertices between each vertex pair. The algorithm iterates through the vertices, using dynamic programming to update the distance matrix and find shorter paths through intermediates at each step. It runs in O(V3) time, where V is the number of vertices.
The document discusses set theory and relations. It defines sets and subsets, set operations including union, intersection, and complement, and properties of sets like cardinality and power sets. Examples are provided to demonstrate counting elements in sets and using Venn diagrams to represent relationships between sets.
- The document explores Bezier curves with quadratic control point interpolations rather than linear interpolations.
- For a Bezier curve with n original control points and quadratic interpolations between points, the degree of the resulting curve is 2 times the degree of a standard Bezier curve with n points and linear interpolations.
- This is because the quadratic interpolations can be viewed as individual Bezier curves themselves, composed together to form the overall curve.
There is only one geometrical method called Exhaustion Method to find out the length of the circumference of a circle. In this method, a regular polygon of known number and value of sides is inscribed, doubled many times until the inscribed polygon exhausts the space between the polygon and circle as limit. In the present paper, it is made clear, that the value for circumference, i.e. 3.14159265358…. of polygon, attributing to circle is a lower value than the real value, and the real value is 14 24 = 3.14644660942 adopting error-free method.
This paper investigates fuzzy triangle and similarity of fuzzy triangles. Five rules to determine similarity of
fuzzy triangles are studied. Extension principle and concept of same and inverse points in fuzzy geometry
are used to analyze the proposed concepts.
This document provides an introduction to probability theory. It begins by defining probability spaces, which consist of a sample space (Ω), a σ-algebra of events (F), and a probability measure (P). It then discusses examples of σ-algebras and probability measures. The document outlines the topics that will be covered, including random variables, integration theory, distributions, and limit theorems. It presents probability theory using a rigorous mathematical approach based on measure theory and σ-algebras.
The document discusses closures of relations, including reflexive closure and symmetric closure. It provides definitions and theorems related to closures. It also uses an example to illustrate finding the reflexive closure and symmetric closure of a relation. Additionally, it covers topics like paths in directed graphs, shortest paths, and transitive closure. It includes an example of calculating the transitive closure of a relation by finding its zero-one matrix.
This document provides an introduction to vectors using a geometric approach. It begins by defining vectors as oriented line segments representing displacements, velocities, and forces. Key concepts introduced include vector addition and scalar multiplication. These operations are used to define a vector space, which has properties like closure under addition and scalar multiplication. Specific vector spaces discussed include Rn, the set of n-tuples of real numbers, and Cn, the set of n-tuples of complex numbers. The document also covers bases, linear independence, components of vectors with respect to a basis, and the dimension of a vector space. Several exercises are provided to reinforce these concepts.
Representing Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 14, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
The document presents the bisection method for finding the root of a continuous function within a given interval. It explains that the bisection method systematically narrows the interval by calculating the midpoint and determining if the root lies in the upper or lower half. It provides pseudocode for implementing the bisection method to find the square root of a number by iteratively calculating the midpoint of the interval until the difference is within a tolerance value.
This document provides an overview of set theory including definitions of key terms and concepts. It defines a set as a well-defined collection of objects or elements. It discusses set operations like union, intersection, difference, and Cartesian product. It defines countable and uncountable sets and proves results like Cantor's theorem, which states that the cardinality of a power set is greater than the original set. Formal proofs involving sets are also covered.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses parametric equations of lines. It explains that a line L passing through a point P = (c, d) in the direction of a vector D = <a, b> can be defined parametrically as L = P + tD, where t is a scalar. This vector equation can be written as parametric equations x(t) = at + c and y(t) = bt + d, where (x(t), y(t)) defines all points on the line L. An example shows how to find the parametric equations for a line passing through a given point in a given direction.
This document provides an overview of key concepts in vector calculus, including:
- Vector calculus concepts such as limits, continuity, derivatives, and integrals of vector functions.
- Operations on vectors like addition, subtraction, scalar multiplication.
- Vector products including the dot product and cross product.
- Applications of vector calculus like determining velocity, acceleration, and line, surface, and volume integrals.
- Parametric representations of curves using a parameter like time, and their relation to geometry.
The document defines sequences and series, and describes three types of progressions: arithmetic, geometric, and harmonic. It then focuses on arithmetic progressions (AP), defining them as sequences where the difference between consecutive terms is constant. The nth term of an AP is given as an + (n-1)d, where a is the first term and d is the common difference. The sum of the first n terms of an AP is provided as (2a + (n-1)d)/2. Arithmetic means are also discussed, along with finding the number of arithmetic means between two numbers in an AP.
The document defines sequences and series, and describes three types of progressions: arithmetic, geometric, and harmonic. It then focuses on arithmetic progressions (AP), defining them as sequences where the difference between consecutive terms is constant. The nth term of an AP is given as an + (n-1)d, where a is the first term and d is the common difference. The sum of the first n terms of an AP is provided as (2a + (n-1)d)/2. Arithmetic means are also discussed, along with finding the number of arithmetic means between two numbers in an AP.
Vectors can be represented as arrows with a magnitude (length) and direction. They are used in mathematics and physics to represent numerical measurements in a specified direction. There are rules for vector addition and scalar multiplication that involve combining or extending/compressing vectors. Vectors can be added using the parallelogram rule or the base-to-tip rule, with the sum being independent of order for three or more vectors.
This document provides information about sequences and series in mathematics. It defines a sequence as a function whose domain is the set of natural numbers and whose range is a set of term values. Examples of finding the next term in sequences are provided. Summation notation is introduced to write the terms of a series and evaluate its sum. Convergent and divergent sequences are defined. Properties of sequences and summation are outlined, including using Desmos to list terms of a sequence. Examples are provided to demonstrate evaluating finite and infinite series using summation notation and properties. Exercises for students are listed at the end.
Squares have the same meaning as quadratic and refer to something that is multiplied by itself or involves terms of the second degree. When you square a number, it is multiplied by itself to give a positive result. A square is also a specific geometric shape with four equal sides that meet at right angles. The area of a square can be calculated by multiplying the length of its sides or taking half of the diagonal squared, while the perimeter is four times the length of its sides.
Vectors are numerical measurements of direction and magnitude. They are represented geometrically by arrows, with the length indicating magnitude and direction pointing the tip. Vector addition follows the parallelogram rule, where the resultant vector is the diagonal of the parallelogram formed by the two vectors. It can also be done by the base to tip rule, placing the base of one vector at the tip of the other. Scalar multiplication scales the vector by a numerical factor, extending or compressing its length.
The Floyd-Warshall algorithm finds the shortest paths between all pairs of vertices in a weighted graph. It does this by considering all possible paths through intermediate vertices between each vertex pair. The algorithm iterates through the vertices, using dynamic programming to update the distance matrix and find shorter paths through intermediates at each step. It runs in O(V3) time, where V is the number of vertices.
The document discusses set theory and relations. It defines sets and subsets, set operations including union, intersection, and complement, and properties of sets like cardinality and power sets. Examples are provided to demonstrate counting elements in sets and using Venn diagrams to represent relationships between sets.
- The document explores Bezier curves with quadratic control point interpolations rather than linear interpolations.
- For a Bezier curve with n original control points and quadratic interpolations between points, the degree of the resulting curve is 2 times the degree of a standard Bezier curve with n points and linear interpolations.
- This is because the quadratic interpolations can be viewed as individual Bezier curves themselves, composed together to form the overall curve.
There is only one geometrical method called Exhaustion Method to find out the length of the circumference of a circle. In this method, a regular polygon of known number and value of sides is inscribed, doubled many times until the inscribed polygon exhausts the space between the polygon and circle as limit. In the present paper, it is made clear, that the value for circumference, i.e. 3.14159265358…. of polygon, attributing to circle is a lower value than the real value, and the real value is 14 24 = 3.14644660942 adopting error-free method.
This paper investigates fuzzy triangle and similarity of fuzzy triangles. Five rules to determine similarity of
fuzzy triangles are studied. Extension principle and concept of same and inverse points in fuzzy geometry
are used to analyze the proposed concepts.
This document provides an introduction to probability theory. It begins by defining probability spaces, which consist of a sample space (Ω), a σ-algebra of events (F), and a probability measure (P). It then discusses examples of σ-algebras and probability measures. The document outlines the topics that will be covered, including random variables, integration theory, distributions, and limit theorems. It presents probability theory using a rigorous mathematical approach based on measure theory and σ-algebras.
The document discusses closures of relations, including reflexive closure and symmetric closure. It provides definitions and theorems related to closures. It also uses an example to illustrate finding the reflexive closure and symmetric closure of a relation. Additionally, it covers topics like paths in directed graphs, shortest paths, and transitive closure. It includes an example of calculating the transitive closure of a relation by finding its zero-one matrix.
This document provides an introduction to vectors using a geometric approach. It begins by defining vectors as oriented line segments representing displacements, velocities, and forces. Key concepts introduced include vector addition and scalar multiplication. These operations are used to define a vector space, which has properties like closure under addition and scalar multiplication. Specific vector spaces discussed include Rn, the set of n-tuples of real numbers, and Cn, the set of n-tuples of complex numbers. The document also covers bases, linear independence, components of vectors with respect to a basis, and the dimension of a vector space. Several exercises are provided to reinforce these concepts.
Representing Relations
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 14, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
CMSC 56 | Lecture 16: Equivalence of Relations & Partial Orderingallyn joy calcaben
Equivalence of Relations & Partial Ordering
CMSC 56 | Discrete Mathematical Structure for Computer Science
November 21, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
The document presents the bisection method for finding the root of a continuous function within a given interval. It explains that the bisection method systematically narrows the interval by calculating the midpoint and determining if the root lies in the upper or lower half. It provides pseudocode for implementing the bisection method to find the square root of a number by iteratively calculating the midpoint of the interval until the difference is within a tolerance value.
This document provides an overview of set theory including definitions of key terms and concepts. It defines a set as a well-defined collection of objects or elements. It discusses set operations like union, intersection, difference, and Cartesian product. It defines countable and uncountable sets and proves results like Cantor's theorem, which states that the cardinality of a power set is greater than the original set. Formal proofs involving sets are also covered.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
The document discusses parametric equations of lines. It explains that a line L passing through a point P = (c, d) in the direction of a vector D = <a, b> can be defined parametrically as L = P + tD, where t is a scalar. This vector equation can be written as parametric equations x(t) = at + c and y(t) = bt + d, where (x(t), y(t)) defines all points on the line L. An example shows how to find the parametric equations for a line passing through a given point in a given direction.
This document provides an overview of key concepts in vector calculus, including:
- Vector calculus concepts such as limits, continuity, derivatives, and integrals of vector functions.
- Operations on vectors like addition, subtraction, scalar multiplication.
- Vector products including the dot product and cross product.
- Applications of vector calculus like determining velocity, acceleration, and line, surface, and volume integrals.
- Parametric representations of curves using a parameter like time, and their relation to geometry.
The document discusses lines and planes in 3D space. It defines lines as being determined by a point and direction vector, and gives parametric and symmetric equations to represent lines. Planes are defined by a point and normal vector, with standard and general forms for their equations. Methods are provided for finding the intersection of lines or planes, as well as the distance between a point and plane or line. Examples demonstrate finding equations of lines and planes, sketching planes, and determining relationships between lines or planes.
This document provides solutions to logistics management assignment problems. It addresses 6 problems related to network flows, facility location, and vehicle routing. For problem 1, it shows that the difference between the sum of outdegrees and indegrees in a network is always zero. For problem 2, it proves Goldman's majority theorem and provides an algorithm to solve the 1-median problem. The solutions utilize concepts like isthmus edges, node weights, and network separation.
This document defines key concepts related to lines in the Euclidean plane including:
1. The definition of a line L as the set of points P0 + ta, where P0 is a base point, a is a non-zero direction vector, and t is a real parameter.
2. Methods for finding the equation of a line including the vector form, parametric form, symmetric form, normal form, and point-slope form.
3. Concepts such as the angle of inclination and slope of a line, and conditions for parallelism and orthogonality between lines.
The document provides an introduction to tensors and their properties. It discusses:
1) Tensors map vectors to other vectors and can be represented by matrices. Common examples are the stress tensor and deformation gradient tensor.
2) Operations performed on matrices, such as addition, multiplication, and transposition, can also be performed on tensors when expressed in terms of their components.
3) Tensors differ from ordinary matrices in that their components must transform in a certain way under a change of basis to maintain the tensor's physical meaning.
1. Vectors have both magnitude and direction, while scalars only have magnitude.
2. Common vector quantities include velocity and force, while common scalars include mass and time.
3. Vectors can be represented by arrows in diagrams or with signs to indicate direction in equations. The resultant vector represents the total effect of multiple vectors.
Chapter 4: Vector Spaces - Part 1/Slides By PearsonChaimae Baroudi
This document defines vectors and vector spaces. It begins by defining vectors in 2D and 3D space as matrices and describes operations like addition, scalar multiplication, and subtraction. It then defines a vector space as a set of vectors that satisfies 10 axioms related to these operations. Examples of vector spaces include the set of 2D and 3D vectors, sets of matrices, and sets of polynomials. The document also defines subspaces and proves that the span of a set of vectors in a vector space forms a subspace.
This document discusses three types of vectors: numeric vectors, geometric/physical vectors, and functions. Numeric vectors are lists of numbers. Geometric/physical vectors have magnitude and direction, like directed line segments representing displacements. Functions can also be viewed as vectors. All three types of vectors can be added, subtracted, and multiplied by numbers. Numeric vectors correspond to geometric vectors through their components in a coordinate system. Forces are represented as geometric vectors with magnitude and direction.
- The binormal vector B(t) is defined as the cross product of the unit tangent vector T(t) and unit normal vector N(t).
- It is proven that B(t) is a unit vector, meaning it has constant length. Its derivative dB/ds is therefore orthogonal to B(t).
- The torsion τ of a space curve is defined as the rate of change of the binormal vector with respect to arc length s, or τ = -dB/ds·N. Torsion measures how much a curve twists as one moves along it.
- For a plane curve, the torsion is always zero since the cross product that defines torsion is equal to the
This document discusses vectors and their components, magnitude, direction, and addition. It provides examples of calculating the magnitude and component form of vectors. It also explains how to add vectors using the parallelogram method by drawing a parallelogram and finding the diagonal vector sum, or the component method by finding the horizontal and vertical components and adding them. Vector addition is important in physics for the law of conservation of momentum.
The document defines a vector as having both magnitude and direction, represented geometrically by an arrow. It discusses representing vectors algebraically using coordinates, and defines operations like addition, subtraction, and scaling of vectors. Key vector concepts covered include the dot product, which yields a scalar when combining two vectors, and unit vectors, which have a magnitude of 1. Examples are provided of using vectors to solve problems and prove geometric properties.
The document defines vectors and discusses their geometric and algebraic representations. Geometrically, a vector has a magnitude and direction represented by an arrow. Algebraically, a vector in a plane can be represented by its coordinates (a1, a2) and in 3D space by coordinates (a1, a2, a3). Vectors can be added by placing them head to tail, subtracted by reversing one and adding, and scaled by a scalar number. The dot product of two vectors A and B yields a scalar value that geometrically equals the magnitudes of A and B multiplied by the cosine of the angle between them.
The document defines vectors and discusses their geometric and algebraic representations. Geometrically, a vector has a magnitude and direction represented by an arrow. Algebraically, a vector in a plane can be represented by its coordinates (a1, a2) and in 3D space by coordinates (a1, a2, a3). Vectors can be added by placing them head to tail, subtracted by reversing one and adding, and scaled by a scalar number. The dot product of two vectors A and B yields a scalar equal to |A||B|cosθ, where θ is the angle between the vectors.
Vector integration involves integrating a vector field along a curve or path, with the result being a vector quantity. It is similar to scalar integration but deals with vector-valued functions instead of scalar functions. Line integrals calculate the total of a scalar or vector field along a curve and are used in physics and engineering to model quantities depending on the path taken. There are two main types: scalar line integrals and vector line integrals.
Linear algebra concepts like vectors, matrices, and linear transformations are important for recommendation systems. Vectors represent items or users, matrices represent item-user preference data. Linear algebra allows analyzing this data to identify patterns and recommend new items. Key techniques include eigendecomposition to reduce dimensionality and identify important relationships in the data, and singular value decomposition to factor matrices for recommendations. These linear algebra concepts are essential mathematical tools for building personalized recommendation models.
The document discusses different forms of equations for straight lines, including:
- Point-slope form, which defines a line through a known point with a given slope
- Slope-intercept form, which defines a line with a given slope and y-intercept
- Normal form, which defines a line based on its perpendicular distance from the origin and the angle it forms with the x-axis
- General form, which is the standard Ax + By + C = 0 equation for a line
It also covers how to find the distance from a point to a line, and how parallel and perpendicular lines can be identified based on the coefficients in their equations.
This document discusses different types of equations for geometric shapes and concepts in the coordinate plane. It provides equations for lines, including vector and parametric forms. It also gives the equations for circles, parabolas, ellipses, hyperbolas, and conic sections. Examples are included for finding the midpoint between two points and the distance between two points. The document serves as a reference for the key equations involved in analytic geometry.
This document provides an overview of mathematical preliminaries including sets, functions, relations, graphs, and proof techniques. It defines sets, set operations, and set representations. It also covers functions, relations, Cartesian products, graphs including walks, paths, cycles, and trees. Finally, it discusses proof techniques like induction and contradiction and provides examples of proofs using these techniques.
Similar to 267 5 parametric eequations of lines (20)
The document introduces matrices and matrix operations. Matrices are rectangular tables of numbers that are used for applications beyond solving systems of equations. Matrix notation defines a matrix with R rows and C columns as an R x C matrix. The entry in the ith row and jth column is denoted as aij. Matrices can be added or subtracted if they are the same size by adding or subtracting the corresponding entries. There are two types of matrix multiplication: scalar multiplication multiplies a matrix by a constant, and matrix multiplication involves multiplying corresponding rows and columns where the number of columns of the left matrix equals the rows of the right matrix.
35 Special Cases System of Linear Equations-x.pptxmath260
The document discusses special cases of systems of linear equations, including inconsistent/contradictory systems where the equations are impossible to satisfy simultaneously, and dependent systems where there are infinitely many solutions. An inconsistent system is shown with equations x + y = 2 and x + y = 3, which has no solution since they cannot both be true. A dependent system is shown with equations x + y = 2 and 2x + 2y = 4, which has infinitely many solutions like (2,0) and (1,1). The row-reduced echelon form (rref) of a matrix is also discussed, which puts a system of equations in a standard form to help determine if it is consistent, dependent, or has
The document discusses conic sections and ellipses. Conic sections are graphs of quadratic equations of the form Ax2 + By2 + Cx + Dy = E, where A and B are not both 0. Their graphs include circles, ellipses, parabolas and hyperbolas. Ellipses are defined as the set of all points where the sum of the distances to two fixed foci is a constant. Ellipses have a center, two axes called the semi-major and semi-minor axes, and radii along the x and y axes called the x-radius and y-radius. The standard form of an ellipse equation is presented.
This document discusses first degree functions and linear equations. It explains that most real-world mathematical functions can be composed of algebraic, trigonometric, or exponential/log formulas. Linear equations of the form Ax + By = C represent straight lines that can be graphed by finding the x- and y-intercepts. If an equation contains only one variable, it represents a vertical or horizontal line. The slope-intercept form y = mx + b is introduced, where m is the slope and b is the y-intercept. Slope is defined as the ratio of the rise over the run between two points on a line.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Most mathematical functions are represented by formulas like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input variable, and the formula defines the relationship between input and output. New functions can be formed using basic operations like addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate evaluating functions at given inputs and combining functions algebraically.
The document discusses exponents and exponent rules. It defines exponents as the number of times a base is multiplied by 1. It presents rules for multiplying, dividing, and raising exponents. Examples are provided to demonstrate applying the rules, such as using the power-multiply rule to evaluate (22*34)3. Special exponent rules are also covered, such as the 0-power rule where A0 equals 1 when A is not 0. The document provides examples of calculating fractional exponents by first extracting the root and then raising it to the numerator power.
The document discusses functions and their basic language. It defines a function as a procedure that assigns each input exactly one output. It provides examples of functions, such as a license number to name function. It explains that a function must have a domain (set of inputs) and range (set of outputs). Functions can be represented graphically, through tables of inputs and outputs, or with mathematical formulas.
The document discusses using sign charts to solve polynomial and rational inequalities. It provides examples of solving inequalities by setting one side equal to zero, factoring the expression, drawing the sign chart, and determining the solutions from the regions with the appropriate signs. Specifically, it works through examples of solving x^2 - 3x > 4, 2x^2 - x^3/(x^2 - 2x + 1) < 0, and (x - 2)/(2/(x - 1)) < 3.
The document discusses sign charts for factorable formulas. It provides examples of determining the sign (positive or negative) of expressions when evaluated at given values of x by factoring the expressions into their factored forms. The key steps to create a sign chart are: 1) solve for values where the expression is equal to 0, 2) mark these values on a number line, 3) select points in each segment to test the sign, 4) indicate the sign (positive or negative) in each segment based on the testing. Sign charts show the regions where an expression is positive, negative or equal to 0.
19 more parabolas a& hyperbolas (optional) xmath260
After dividing the general quadratic equation Ax2 + By2 + Cx + Dy = E by A, three types of conic sections can be obtained:
1) Parabolas occur when B = 0, resulting in equations of the form 1x2 + #x + #y = #.
2) Circles occur when A = B, resulting in the equation 1x2 + 1y2 = 1.
3) Hyperbolas occur when A and B have opposite signs, resulting in equations of the form 1x2 + ry2 + #x + #y = # with r < 0. Hyperbolas have two foci and asymptotes, and points on the hyperbola have
The document discusses conic sections, specifically circles and ellipses. It defines an ellipse as the set of points where the sum of the distances to two fixed foci is a constant. An ellipse has a center, two axes (semi-major and semi-minor), and can be represented by the standard form (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. Examples are provided to demonstrate finding attributes of ellipses from their equations.
This document discusses conic sections and first degree equations. It begins by introducing conic sections as the shapes formed by slicing a cone at different angles. It then covers first degree equations, noting that their graphs are straight lines that can be written in the form of y=mx+b. Specific examples of first degree equations and their graphs are shown. The document ends by introducing the four types of conic sections - circles, ellipses, parabolas, and hyperbolas - and how graphs of second degree equations can represent these shapes.
The document discusses calculating the slope of a curve between two points (x, f(x)) and (x+h, f(x+h)) using the difference quotient formula. It defines the difference quotient as (f(x+h) - f(x))/h, where h is the difference between x and x+h. An example calculates the slope between the points (2, f(2)) and (2.2, f(2.2)) for the function f(x) = x^2 - 2x + 2, finding the slope to be 0.44.
The document discusses various transformations that can be performed on graphs of functions, including vertical translations, stretches, and compressions. Vertical translations move the entire graph up or down by adding or subtracting a constant to the function. Stretches elongate or compress the graph vertically by multiplying the function by a constant greater than or less than 1, respectively. These transformations can be represented by modifying the original function in a way that corresponds to the geometric transformation of its graph.
14 graphs of factorable rational functions xmath260
The document discusses graphs of rational functions. It defines rational functions as functions of the form R(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials. It describes how vertical asymptotes occur where the denominator Q(x) is zero. The graph runs along either side of vertical asymptotes, going up or down depending on the sign chart. There are four cases for how the graph behaves at a vertical asymptote. The document uses examples to illustrate graphing rational functions and determining vertical asymptotes. It also mentions horizontal asymptotes will be discussed.
The document discusses factorable polynomials and graphing them. It defines a factorable polynomial P(x) as one that can be written as the product of linear factors P(x) = an(x - r1)(x - r2)...(x - rk), where r1, r2, etc. are the roots of P(x). It explains that for large values of |x|, the leading term of P(x) dominates so the graph resembles that of the leading term, while near the roots other terms contribute to the shape of the graph. Examples of graphs of polynomials like x^n are provided to illustrate the approach.
The document discusses quadratic functions and parabolas. It defines quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It states that the graphs of quadratic equations are called parabolas. Parabolas are symmetric around a central line, with the vertex (highest/lowest point) located on this line. The vertex formula is given as x = -b/2a. Steps for graphing a parabola are outlined, including finding the vertex, another point, and reflections across the central line. An example graphs the parabola y = x2 - 4x - 12, finding the vertex as (2, -16) and x-intercepts as -
The document describes the rectangular coordinate system. Each point in a plane can be located using an ordered pair (x,y) where x represents the distance right or left from the origin and y represents the distance up or down. Changing the x-value moves the point right or left, and changing the y-value moves the point up or down. The plane is divided into four quadrants based on the sign of the x and y values. Reflecting a point across an axis results in another point with the same magnitude but opposite sign for the corresponding coordinate.
The document discusses first degree (linear) functions. It states that most real-world mathematical functions can be composed of formulas from three families: algebraic, trigonometric, and exponential-logarithmic. It focuses on linear functions of the form f(x)=mx+b, where m is the slope and b is the y-intercept. Examples are given of equations and how to determine the slope and y-intercept to write the equation in slope-intercept form as a linear function.
The document discusses the basic language of functions. It defines a function as a procedure that assigns each input exactly one output. Functions can be represented by formulas using typical variables like f(x) = x^2 - 2x + 3, where x is the input and f(x) is the output. Functions have a domain, which is the set of all possible inputs, and a range, which is the set of all possible outputs. Functions can be depicted graphically or via tables listing inputs and outputs.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
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How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
3. Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Parametric Equations of Lines
4. Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ.
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Parametric Equations of Lines
5. Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Parametric Equations of Lines
6. Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
–2v –v
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Parametric Equations of Lines
7. Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
–2v –v
The Base to Tip Rule
Given two vectors u and v, u + v is the
new vector formed by placing the base
of one vector at the tip of the other.
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Parametric Equations of Lines
8. Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
–2v –v
The Base to Tip Rule
Given two vectors u and v, u + v is the
new vector formed by placing the base
of one vector at the tip of the other.
u
v
The Base to Tip Rule
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Parametric Equations of Lines
9. Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
–2v –v
The Base to Tip Rule
Given two vectors u and v, u + v is the
new vector formed by placing the base
of one vector at the tip of the other.
v
u
v
The Base to Tip Rule
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Parametric Equations of Lines
10. Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
–2v –v
The Base to Tip Rule
Given two vectors u and v, u + v is the
new vector formed by placing the base
of one vector at the tip of the other.
v
u
u + v
v
The Base to Tip Rule
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Parametric Equations of Lines
11. Scalar Multiplication
Given a number λ and a vector v,
λv is the extension (compression)
of the vector v by a factor λ. v 2v
–2v –v
The Base to Tip Rule
Given two vectors u and v, u + v is the
new vector formed by placing the base
of one vector at the tip of the other.
v
u
u + v
v
The Base to Tip Rule
Equations of Lines
We start with a vector motivated representation of
lines by recalling the following associations.
Again, we begin with the 2D version
then generalize the results to 3D.
Parametric Equations of Lines
12. Let D = <a, b> be a vector that indicates direction as
shown.
D=<a,b>
Parametric Equations of Lines
x
13. Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D.
D=<a,b>
tD=<ta,tb>
Parametric Equations of Lines
x
14. we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
D=<a,b>
tD=<ta,tb>
Parametric Equations of Lines
x
15. Hence x(t) = at, y(t) = bt
is a set of parametric equations
for the line through the origin in
the direction of D = <a, b>.
D=<a,b>
tD=<ta,tb>
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
Parametric Equations of Lines
x
16. Hence x(t) = at, y(t) = bt
is a set of parametric equations
for the line through the origin in
the direction of D = <a, b>.
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
D=<a,b>
tD=<ta,tb>
P=<c, d>
Suppose P = (c, d) specifies a
base point,
Parametric Equations of Lines
x
17. Hence x(t) = at, y(t) = bt
is a set of parametric equations
for the line through the origin in
the direction of D = <a, b>.
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
Suppose P = (c, d) specifies a
base point, using the base-to-tip
addition rule, the vector sums
P + tD = <c, d> + t<a, b>
D=<a,b>
tD=<ta,tb>
P=<c, d>
Parametric Equations of Lines
x
18. Hence x(t) = at, y(t) = bt
is a set of parametric equations
for the line through the origin in
the direction of D = <a, b>.
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
D=<a,b>
tD=<ta,tb>
P=<c, d>
Suppose P = (c, d) specifies a
base point, using the base-to-tip
addition rule, the vector sums
P + tD = <c, d> + t<a, b>
Parametric Equations of Lines
x
19. Hence x(t) = at, y(t) = bt
is a set of parametric equations
for the line through the origin in
the direction of D = <a, b>.
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
Suppose P = (c, d) specifies a
base point, using the base-to-tip
addition rule, the vector sums
P + tD = <c, d> + t<a, b>
Parametric Equations of Lines
x
20. Hence x(t) = at, y(t) = bt
is a set of parametric equations
for the line through the origin in
the direction of D = <a, b>.
we get all the points in the "direction" of D,
i.e. the entire line that coincides with D.
Let D = <a, b> be a vector that indicates direction as
shown. Let t be a scalar, then t*D = <ta, tb> is an
extension of D. As t takes on all real numbers,
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
Suppose P = (c, d) specifies a
base point, using the base-to-tip
addition rule, the vector sums
P + tD = <c, d> + t<a, b> = <c + ta, d + tb>
viewed as points, are all the points on the line L.
L
Parametric Equations of Lines
x
21. D=<a,b>P=<c, d>
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b>
Parametric Equations of Lines
x
22. D=<a,b>P=<c, d>
L
To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b>
Parametric Equations of Lines
x
23. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Parametric Equations of Lines
D=<a,b>P=<c, d>
L
x
24. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Parametric Equations of Lines
D=<a,b>
tD=<ta,tb>
P=<c, d>
L
x
25. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Parametric Equations of Lines
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
L
x
26. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Parametric Equations of Lines
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
L
x
27. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Parametric Equations of Lines
This is a vector–equation of L.
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
L
x
28. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Parametric Equations of Lines
If (x, y) is a generic point on L then
(x, y) = (c, d) + t*(a, b) = (c + at, d + bt).
This is a vector–equation of L.
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
L
x
29. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Hence x = c + at, y = d + bt is a set of
parametric equations of L.
Parametric Equations of Lines
If (x, y) is a generic point on L then
(x, y) = (c, d) + t*(a, b) = (c + at, d + bt).
This is a vector–equation of L.
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
L
x
30. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Hence x = c + at, y = d + bt is a set of
parametric equations of L.
Example A. Find a set of
parametric equations for the line L
a. in the direction of D = <2, 1>,
passing through P = (4, 5).
Parametric Equations of Lines
If (x, y) is a generic point on L then
(x, y) = (c, d) + t*(a, b) = (c + at, d + bt).
This is a vector–equation of L.
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
L
x
31. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Hence x = c + at, y = d + bt is a set of
parametric equations of L.
Example A. Find a set of
parametric equations for the line L
a. in the direction of D = <2, 1>,
passing through P = (4, 5). P=(4, 5)
D=<2, 1>
x
Parametric Equations of Lines
If (x, y) is a generic point on L then
(x, y) = (c, d) + t*(a, b) = (c + at, d + bt).
This is a vector–equation of L.
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
L
x
32. To summarize, let L be the line passing though the
point P = (c, d) in the direction of D = <a, b> then
L = P+ t*D where t is a number.
Hence x = c + at, y = d + bt is a set of
parametric equations of L.
Example A. Find a set of
parametric equations for the line L
a. in the direction of D = <2, 1>,
passing through P = (4, 5).
The parametric equations of L are
x(t) = 2t + 4, y(t) = 1t + 5.
P=(4, 5)
D=<2, 1>
( 2t + 4, 1t + 5)
x
Parametric Equations of Lines
If (x, y) is a generic point on L then
(x, y) = (c, d) + t*(a, b) = (c + at, d + bt).
This is a vector–equation of L.
D=<a,b>
tD=<ta,tb>
P=<c, d>
<at+c,bt+d>
L
x
33. the set of points tD =(at, bt, ct)
form the line through the origin
coinciding with the vector D or
that it’s in the same direction as D.
Parametric Equations of Lines
Let D = <a, b, c> be a vector as shown,
Line Equations R3
34. the set of points tD =(at, bt, ct)
form the line through the origin
coinciding with the vector D or that
it’s in the same direction as D. tD=<at,bt,ct>
D = <a, b, c>
Parametric Equations of Lines
x
y
z
Let D = <a, b, c> be a vector as shown,
Line Equations R3
35. the set of points tD =(at, bt, ct)
form the line through the origin
coinciding with the vector D or that
it’s in the same direction as D.
Let P = (d, e, f) be a point in space,
tD=<at,bt,ct>
P=(d,e,f)
D = <a, b, c>
Parametric Equations of Lines
x
y
z
Let D = <a, b, c> be a vector as shown.
Line Equations R3
36. the set of points tD =(at, bt, ct)
form the line through the origin
coinciding with the vector D or that
it’s in the same direction as D.
Let P = (d, e, f) be a point in space,
tD=<at,bt,ct>
P=(d,e,f)
D = <a, b, c>
of points P + tD = (d + at, e + bt, f + ct ).
P+tD
Parametric Equations of Lines
x
y
then the line containing P,
in the direction of D consists
z
Let D = <a, b, c> be a vector as shown.
Line Equations R3
37. the set of points tD =(at, bt, ct)
form the line through the origin
coinciding with the vector D or that
it’s in the same direction as D.
Let P = (d, e, f) be a point in space,
tD=<at,bt,ct>
Hence x(t) = at + d, y(t) = bt + e, z(t) = ct + f,
where t is any number, is a set of parametric
equations representing the line through
P = (d, e, f) in the direction D = <a, b, c>.
P=(d,e,f)
D = <a, b, c>
of points P + tD = (d + at, e + bt, f + ct ).
P+tD
Parametric Equations of Lines
x
y
then the line containing P,
in the direction of D consists
z
Let D = <a, b, c> be a vector as shown.
Line Equations R3
38. Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
Parametric Equations of Lines
39. Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
Q(3,–2,5)
x
y
z
R(1,2,1)
QR=D
Parametric Equations of Lines
40. Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
The vector QR = R – Q
= <1, 2, 1> – <3, –2, 5>
= <–2, 4, –4> = D is a
vector that gives the
direction of the line.
Parametric Equations of Lines
Q(3,–2,5)
x
y
z
R(1,2,1)
QR=D
41. Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
The vector QR = R – Q
= <1, 2, 1> – <3, –2, 5>
= <–2, 4, –4> = D is a
vector that gives the
direction of the line. So
x(t) = –2t + 4
y(t) = 4t + 5
z(t) = –4t + 6
Parametric Equations of Lines
is the line through P that has the same direction as QR.
Q(3,–2,5)
x
y
z
R(1,2,1)
QR=D
42. Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
The vector QR = R – Q
= <1, 2, 1> – <3, –2, 5>
= <–2, 4, –4> = D is a
vector that gives the
direction of the line. So
x(t) = –2t + 4
y(t) = 4t + 5
z(t) = –4t + 6
Q(3,–2,5)
x
y
z
R(1,2,1)
P(4,5,6)
QR=D
P+t*QR
Parametric Equations of Lines
is the line through P that has the same direction as QR.
43. Example B. Find a set of parametric equations for the
line in the same direction as QR where Q = (3, –2, 5),
R = (1, 2, 1) that passes through P = (4, 5, 6).
The vector QR = R – Q
= <1, 2, 1> – <3, –2, 5>
= <–2, 4, –4> = D is a
vector that gives the
direction of the line. So
x(t) = –2t + 4
y(t) = 4t + 5
z(t) = –4t + 6
Q(3,–2,5)
x
y
z
R(1,2,1)
P(4,5,6)
QR=D
P+t*QR
Parametric Equations of Lines
is the line through P that has the same direction as QR.
Definition: Two lines are parallel if they have the same
(or opposite) directional vector.
44. Let L represent the line in the last example, then we
may write L = P + t*D where D is a vector that gives
the direction of the line, P is a vector whose tip is on
the line L, and t as the variable. This is called the
vector-equation of the line.
Q
x
y
z
R
P
QR = D
L = P+t*D
Parametric Equations of Lines
45. Let L represent the line in the last example, then we
may write L = P + t*D where D is a vector that gives
the direction of the line, P is a vector whose tip is on
the line L, and t as the variable. This is called the
vector-equation of the line.
Q
x
y
z
R
P
QR = D
L = P+t*DRemark: We can't
represent a line in 3D with
a single equation in the
variable x, y and z
because the graph of such
an equation is a surface in
3D space in general.
Parametric Equations of Lines
46. Let L represent the line in the last example, then we
may write L = P + t*D where D is a vector that gives
the direction of the line, P is a vector whose tip is on
the line L, and t as the variable. This is called the
vector-equation of the line.
Q
x
y
z
R
P
QR = D
L = P+t*DRemark: We can't
represent a line in 3D with
a single equation in the
variable x, y and z
because the graph of such
an equation is a surface in
3D space in general.
Parametric Equations of Lines
Another method of setting equations to represent a
line L is to give L as the intersection of two planes.
47. Let L be the line <t, t, t > where t is any real number,
as shown here.
Parametric Equations of Lines
x
z+
<1,1,1>
y
48. Let L be the line <t, t, t > where t is any real number,
as shown here.
Parametric Equations of Lines
x
z+
<1,1,1>
y
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
49. Let L be the line <t, t, t > where t is any real number,
as shown here.
or that (t =) x = y = z.
Parametric Equations of Lines
x
z+
<1,1,1>
y
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
50. Let L be the line <t, t, t > where t is any real number,
as shown here.
or that (t =) x = y = z.
These triple–equation is called the
symmetric equation of the line L.
Parametric Equations of Lines
x
z+
<1,1,1>
y
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
51. Let L be the line <t, t, t > where t is any real number,
as shown here.
or that (t =) x = y = z.
These triple–equation is called the
symmetric equation of the line L.
Parametric Equations of Lines
x
z+
<1,1,1>
y
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
The symmetric equation actually consists of two
systems of linear equations,
52. Let L be the line <t, t, t > where t is any real number,
as shown here.
or that (t =) x = y = z.
These triple–equation is called the
symmetric equation of the line L.
Parametric Equations of Lines
x
z+
<1,1,1>
y
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
The symmetric equation actually consists of two
systems of linear equations, in this case
x = y
y = z
x = y
x = z
x = z
y = zA: B: C:
53. Let L be the line <t, t, t > where t is any real number,
as shown here.
or that (t =) x = y = z.
These triple–equation is called the
symmetric equation of the line L.
Parametric Equations of Lines
x
z+
<1,1,1>
y
The parametric equations for L are
x(t) = t, y(t) = t, z(t) = t
The symmetric equation actually consists of two
systems of linear equations, in this case
x = y
y = z
x = y
x = z
x = z
y = zA: B: C:
Each system of equations consist of two planes and
L is the intersection of two planes.