This document provides information about 3-D vectors including:
1. Basic concepts of vectors such as notation, characteristics, addition and subtraction laws.
2. Components of vectors in 2-D and 3-D space including position vectors.
3. Dot/scalar product and its properties including applications to work done.
4. Cross/vector product and its properties including the determinant formula and applications.
The document provides solutions to exercises on designing context-free grammars for various languages. It gives context-free grammars for languages like {anbm | n ≤ m + 3} by partitioning the language into sublanguages that can be generated by nonterminals and combining them. It also explains how to handle more complex cases involving multiple variables like {anbmck | n = m or m ≤ k} by ensuring certain relations between the variables are maintained during derivation.
The document provides an overview of tensor calculus and its notations. It discusses two methods for representing tensors: direct notation which treats tensors as invariant objects, and index notation which uses tensor components. The direct notation is preferred. Basic operations for vectors and second rank tensors are defined, including addition, scalar multiplication, dot products, cross products, and properties. Polar and axial vectors are distinguished. Guidelines are given for tensor calculus notation and rules used throughout the work.
This document introduces vectors and their properties. It defines vectors as quantities that have both magnitude and direction, unlike scalars which only have magnitude. The key concepts covered are:
- Representing vectors geometrically using directed line segments.
- Adding and multiplying vectors, including multiplying a vector by a scalar.
- Using position vectors to represent the location of points relative to a fixed origin.
- Expressing vectors in terms of other known vectors using properties of vector addition and scalar multiplication.
This document provides information about plane and solid geometry. It defines key shapes and formulas for calculating areas and volumes. For plane geometry, it covers triangles, rectangles, squares, quadrilaterals, regular polygons, circles, parabolic and elliptic segments. For solid geometry, it defines polyhedrons, prisms, cylinders, cones, pyramids, spheres, ellipsoids and paraboloids. It provides formulas to calculate properties like areas, volumes, surface areas, circumferences and more for these various geometric shapes.
Slides from a talk called "Projective Geometric Computing given at SIGGRAPH 2000 in New Orleans, 25 July 2000 by Ambjörn Naeve in connection with an advanced course on Geometric Algebra (See http://www.siggraph.org/s2000/conference/courses/crs31.html)
This module discusses solving oblique triangles using the law of sines. It begins by introducing acute and obtuse triangles and how to find the measure of the third angle given two angles. It then derives the law of sines and shows how it can be used to solve triangles where two angles and a side opposite one angle are given, two angles and the included side are given, or all three sides are given. Examples of solving various triangle scenarios are provided.
This document contains lecture notes on mechanics of solids and structures from the University of Manchester. It covers topics related to centroids, moments of area, beams, and bending theory. Specifically, it provides definitions and examples of centroids, first and second moments of area, and introduces beam supports and equilibrium, beam shear forces and bending moments, and bending theory. The contact information for the lecturer, Dr. D.A. Bond, is also provided at the top.
The document discusses the standard form of an equation for a line (Ax + By = C) and proves that any equation in this form represents a line. It provides examples of identifying the slope and y-intercept of lines written in standard form, and notes some advantages of the standard form over the slope-intercept form, such as being able to represent vertical lines. The document concludes by assigning homework problems related to graphing and working with lines in standard form.
The document provides solutions to exercises on designing context-free grammars for various languages. It gives context-free grammars for languages like {anbm | n ≤ m + 3} by partitioning the language into sublanguages that can be generated by nonterminals and combining them. It also explains how to handle more complex cases involving multiple variables like {anbmck | n = m or m ≤ k} by ensuring certain relations between the variables are maintained during derivation.
The document provides an overview of tensor calculus and its notations. It discusses two methods for representing tensors: direct notation which treats tensors as invariant objects, and index notation which uses tensor components. The direct notation is preferred. Basic operations for vectors and second rank tensors are defined, including addition, scalar multiplication, dot products, cross products, and properties. Polar and axial vectors are distinguished. Guidelines are given for tensor calculus notation and rules used throughout the work.
This document introduces vectors and their properties. It defines vectors as quantities that have both magnitude and direction, unlike scalars which only have magnitude. The key concepts covered are:
- Representing vectors geometrically using directed line segments.
- Adding and multiplying vectors, including multiplying a vector by a scalar.
- Using position vectors to represent the location of points relative to a fixed origin.
- Expressing vectors in terms of other known vectors using properties of vector addition and scalar multiplication.
This document provides information about plane and solid geometry. It defines key shapes and formulas for calculating areas and volumes. For plane geometry, it covers triangles, rectangles, squares, quadrilaterals, regular polygons, circles, parabolic and elliptic segments. For solid geometry, it defines polyhedrons, prisms, cylinders, cones, pyramids, spheres, ellipsoids and paraboloids. It provides formulas to calculate properties like areas, volumes, surface areas, circumferences and more for these various geometric shapes.
Slides from a talk called "Projective Geometric Computing given at SIGGRAPH 2000 in New Orleans, 25 July 2000 by Ambjörn Naeve in connection with an advanced course on Geometric Algebra (See http://www.siggraph.org/s2000/conference/courses/crs31.html)
This module discusses solving oblique triangles using the law of sines. It begins by introducing acute and obtuse triangles and how to find the measure of the third angle given two angles. It then derives the law of sines and shows how it can be used to solve triangles where two angles and a side opposite one angle are given, two angles and the included side are given, or all three sides are given. Examples of solving various triangle scenarios are provided.
This document contains lecture notes on mechanics of solids and structures from the University of Manchester. It covers topics related to centroids, moments of area, beams, and bending theory. Specifically, it provides definitions and examples of centroids, first and second moments of area, and introduces beam supports and equilibrium, beam shear forces and bending moments, and bending theory. The contact information for the lecturer, Dr. D.A. Bond, is also provided at the top.
The document discusses the standard form of an equation for a line (Ax + By = C) and proves that any equation in this form represents a line. It provides examples of identifying the slope and y-intercept of lines written in standard form, and notes some advantages of the standard form over the slope-intercept form, such as being able to represent vertical lines. The document concludes by assigning homework problems related to graphing and working with lines in standard form.
This document contains 21 problems solving for the moment of inertia of various shapes. The shapes include rectangles, triangles, semicircles, and composite shapes. For each problem, the relevant dimensions are given, a calculation is shown, and the numerical value of the moment of inertia about the specified axis is provided. Formulas for the moment of inertia of common shapes like rectangles and triangles are used.
Stinespring’s theorem for maps on hilbert c star moduleswtyru1989
This document discusses Stinespring's theorem for completely positive maps on Hilbert C*-modules. It begins by introducing C*-algebras, Hilbert C*-modules, and completely positive maps. It then presents Stinespring's theorem for completely positive maps between C*-algebras. The document goes on to discuss Asadi's generalization of Stinespring's theorem to completely positive maps between a C*-algebra and bounded operators on a Hilbert space that are compatible with a Hilbert C*-module. It concludes by presenting a further generalization of Stinespring's theorem to completely positive maps between a C*-algebra and a Hilbert C*-module.
This document presents an algorithm for finding the minimum local disk cover sets for broadcasting in heterogeneous wireless ad hoc networks. It defines the problem and introduces the concept of a "skyline set" to represent the solution. It then proves several lemmas about the geometry of intersecting disks and uses mathematical induction to show that the number of arcs in any skyline set is upper bounded by 2n, where n is the number of disks. This allows the algorithm to run in O(n log n) time by using a divide-and-conquer approach to merge partial skyline sets.
The document discusses shear forces and bending moments in beams. It provides examples of different types of beams and loads, and how to calculate reactions, shear forces, and bending moments. Equations are given relating loads, shear forces, and bending moments. Methods are described for constructing shear force and bending moment diagrams for beams with different load conditions like concentrated loads, uniform loads, and combinations of loads.
The document contains 10 math problems involving finding equations of lines from graphs, finding gradients, y-intercepts, x-intercepts, and points of intersection of parallel and perpendicular lines. It provides diagrams and step-by-step workings for calculating values related to the straight lines shown. The document tests skills in using properties of straight lines, simultaneous equations, and coordinate geometry.
This document contains notes and formulae on solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, algebraic formulae, linear inequalities, statistics, significant figures and standard form, quadratic expressions and equations, sets, mathematical reasoning, straight lines, and trigonometry. The key concepts covered include formulas for calculating the volume and surface area of various 3D shapes, properties of angles in circles and polygons, factorising and expanding algebraic expressions, solving linear and quadratic equations, set notation and Venn diagrams, types of logical arguments, equations of straight lines, and defining the basic trigonometric ratios.
Module 13 Gradient And Area Under A Graphguestcc333c
1) The document provides examples and questions related to calculating gradient, area under graphs, speed, velocity, and distance from speed-time and distance-time graphs.
2) It includes 10 multi-part questions testing concepts like calculating rate of change of speed, uniform speed, total distance, meeting time, and average speed.
3) Detailed step-by-step answers are provided for each question at the end to demonstrate how to apply the concepts to calculate the requested values.
This document contains instructions and questions for a mathematics preliminary examination. It consists of 7 questions testing skills in algebra, trigonometry, geometry, statistics, and problem solving. Students are instructed to show their working, use formulas provided, and give answers to a specified degree of accuracy. A total of 100 marks are available across the exam.
This document summarizes the numerical solution of the time-independent Schrodinger equation for particles in chaotic stadium and Sinai billiard potentials using the finite difference method. Scars, or regions of high probability density around unstable periodic orbits, were observed in particular eigenstates of both systems, consistent with previous studies. The method was validated by reproducing analytical solutions for 1D and circular wells and showing convergence of eigenvalues with increasing numerical resolution.
This document provides information about polar coordinates including:
- Relations between Cartesian and polar coordinates
- Sketching graphs in polar coordinates such as circles, cardioids, and roses
- Finding intersections of curves, slopes of tangents, and areas bounded by polar curves
- Computing arc lengths and surfaces of revolution generated by polar curves
It discusses key concepts like symmetry properties and provides examples of computing specific values related to polar curves.
Graph theory can be used to represent networks such as oil pipelines, traffic flows, and biochemical pathways. A graph consists of nodes connected by arcs or edges. Simple graphs do not have loops or multiple arcs between node pairs. Complete graphs connect every node to every other node. Subgraphs are formed by removing nodes and/or arcs from a graph. Bipartite graphs have two node sets with no arcs within sets, and complete bipartite graphs connect every node in one set to every node in the other set. Connected graphs allow a path between any two nodes, while Eulerian graphs have all nodes of even degree, allowing a closed trail through every arc once. Semi-Eulerian graphs have exactly two nodes of odd degree
This document discusses segments, angles, and theorems related to them. It introduces the midpoint theorem, which states that if a point M is the midpoint of segment AB, then the distances AM and MB will each be half the length of AB. It also introduces the angle bisector theorem, which states that if BX bisects angle ABC, then the measures of angles ABX and XBC will each be half the measure of ABC. Examples are given to demonstrate applying these theorems. Vertical angles are defined as always being equal in measure. Finally, extra practice problems from 1 to 17 are listed.
The document discusses properties of real numbers, operations on real numbers such as addition, multiplication, and their identities. It also discusses conic sections, their equations and properties. Finally, it covers topics in trigonometry, vectors, complex numbers, and other mathematical functions and formulas.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
The document presents two methods for finding the area of a triangle when the base is known but the perpendicular height is not:
1. Using trigonometry, it derives an expression for the height in terms of one of the angles and the base, leading to the general area formula involving the base, one side, and an opposite angle.
2. Using Pythagorean theorem applied to two triangles, it eliminates the height and derives an expression for the height solely in terms of the triangle's three sides, resulting in Heron's formula for the area.
The unit circle relates real numbers to points on a circle of radius 1 centered at the origin. Each real number t corresponds to a point (x, y) on the circle, allowing the definition of the six trigonometric functions in terms of x and y. The trigonometric functions sine and cosine are periodic with period 2π, meaning their values repeat every 2π units. Their domains are all real numbers and their ranges are between -1 and 1.
IJCER (www.ijceronline.com) International Journal of computational Engineeri...ijceronline
1. The document introduces the concept of a "Total Prime Graph", which is a graph that admits a special type of labeling called a "Total Prime Labeling".
2. Some properties of Total Prime Labelings are studied, and it is proved that paths, stars, bistars, combs, even cycles, helm graphs, and certain wheel graphs are Total Prime Graphs. However, odd cycles are proved to not be Total Prime Graphs.
3. The labeling must satisfy two conditions - the labels of adjacent vertices and incident edges of high degree vertices must be relatively prime. Several examples and theorems demonstrating Total Prime Graphs are provided.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic functions.
This document provides information on differentiation including:
- The definition of the derivative as a limit.
- Rules of differentiation including constant multiples, sums, products, quotients, and chains.
- Derivatives of trigonometric, exponential, and logarithmic functions.
- Examples of calculating derivatives using the definition and rules of differentiation.
This document provides an overview of the Laplace transform and its properties and applications. Specifically, it defines the Laplace transform and inverse Laplace transform, lists several Laplace transform pairs, and provides examples of using the Laplace transform to solve initial value problems involving differential equations with constant coefficients. It also includes a problem set with exercises calculating Laplace transforms and using them to solve initial value problems.
This document contains 21 problems solving for the moment of inertia of various shapes. The shapes include rectangles, triangles, semicircles, and composite shapes. For each problem, the relevant dimensions are given, a calculation is shown, and the numerical value of the moment of inertia about the specified axis is provided. Formulas for the moment of inertia of common shapes like rectangles and triangles are used.
Stinespring’s theorem for maps on hilbert c star moduleswtyru1989
This document discusses Stinespring's theorem for completely positive maps on Hilbert C*-modules. It begins by introducing C*-algebras, Hilbert C*-modules, and completely positive maps. It then presents Stinespring's theorem for completely positive maps between C*-algebras. The document goes on to discuss Asadi's generalization of Stinespring's theorem to completely positive maps between a C*-algebra and bounded operators on a Hilbert space that are compatible with a Hilbert C*-module. It concludes by presenting a further generalization of Stinespring's theorem to completely positive maps between a C*-algebra and a Hilbert C*-module.
This document presents an algorithm for finding the minimum local disk cover sets for broadcasting in heterogeneous wireless ad hoc networks. It defines the problem and introduces the concept of a "skyline set" to represent the solution. It then proves several lemmas about the geometry of intersecting disks and uses mathematical induction to show that the number of arcs in any skyline set is upper bounded by 2n, where n is the number of disks. This allows the algorithm to run in O(n log n) time by using a divide-and-conquer approach to merge partial skyline sets.
The document discusses shear forces and bending moments in beams. It provides examples of different types of beams and loads, and how to calculate reactions, shear forces, and bending moments. Equations are given relating loads, shear forces, and bending moments. Methods are described for constructing shear force and bending moment diagrams for beams with different load conditions like concentrated loads, uniform loads, and combinations of loads.
The document contains 10 math problems involving finding equations of lines from graphs, finding gradients, y-intercepts, x-intercepts, and points of intersection of parallel and perpendicular lines. It provides diagrams and step-by-step workings for calculating values related to the straight lines shown. The document tests skills in using properties of straight lines, simultaneous equations, and coordinate geometry.
This document contains notes and formulae on solid geometry, circle theorems, polygons, factorisation, expansion of algebraic expressions, algebraic formulae, linear inequalities, statistics, significant figures and standard form, quadratic expressions and equations, sets, mathematical reasoning, straight lines, and trigonometry. The key concepts covered include formulas for calculating the volume and surface area of various 3D shapes, properties of angles in circles and polygons, factorising and expanding algebraic expressions, solving linear and quadratic equations, set notation and Venn diagrams, types of logical arguments, equations of straight lines, and defining the basic trigonometric ratios.
Module 13 Gradient And Area Under A Graphguestcc333c
1) The document provides examples and questions related to calculating gradient, area under graphs, speed, velocity, and distance from speed-time and distance-time graphs.
2) It includes 10 multi-part questions testing concepts like calculating rate of change of speed, uniform speed, total distance, meeting time, and average speed.
3) Detailed step-by-step answers are provided for each question at the end to demonstrate how to apply the concepts to calculate the requested values.
This document contains instructions and questions for a mathematics preliminary examination. It consists of 7 questions testing skills in algebra, trigonometry, geometry, statistics, and problem solving. Students are instructed to show their working, use formulas provided, and give answers to a specified degree of accuracy. A total of 100 marks are available across the exam.
This document summarizes the numerical solution of the time-independent Schrodinger equation for particles in chaotic stadium and Sinai billiard potentials using the finite difference method. Scars, or regions of high probability density around unstable periodic orbits, were observed in particular eigenstates of both systems, consistent with previous studies. The method was validated by reproducing analytical solutions for 1D and circular wells and showing convergence of eigenvalues with increasing numerical resolution.
This document provides information about polar coordinates including:
- Relations between Cartesian and polar coordinates
- Sketching graphs in polar coordinates such as circles, cardioids, and roses
- Finding intersections of curves, slopes of tangents, and areas bounded by polar curves
- Computing arc lengths and surfaces of revolution generated by polar curves
It discusses key concepts like symmetry properties and provides examples of computing specific values related to polar curves.
Graph theory can be used to represent networks such as oil pipelines, traffic flows, and biochemical pathways. A graph consists of nodes connected by arcs or edges. Simple graphs do not have loops or multiple arcs between node pairs. Complete graphs connect every node to every other node. Subgraphs are formed by removing nodes and/or arcs from a graph. Bipartite graphs have two node sets with no arcs within sets, and complete bipartite graphs connect every node in one set to every node in the other set. Connected graphs allow a path between any two nodes, while Eulerian graphs have all nodes of even degree, allowing a closed trail through every arc once. Semi-Eulerian graphs have exactly two nodes of odd degree
This document discusses segments, angles, and theorems related to them. It introduces the midpoint theorem, which states that if a point M is the midpoint of segment AB, then the distances AM and MB will each be half the length of AB. It also introduces the angle bisector theorem, which states that if BX bisects angle ABC, then the measures of angles ABX and XBC will each be half the measure of ABC. Examples are given to demonstrate applying these theorems. Vertical angles are defined as always being equal in measure. Finally, extra practice problems from 1 to 17 are listed.
The document discusses properties of real numbers, operations on real numbers such as addition, multiplication, and their identities. It also discusses conic sections, their equations and properties. Finally, it covers topics in trigonometry, vectors, complex numbers, and other mathematical functions and formulas.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
The document presents two methods for finding the area of a triangle when the base is known but the perpendicular height is not:
1. Using trigonometry, it derives an expression for the height in terms of one of the angles and the base, leading to the general area formula involving the base, one side, and an opposite angle.
2. Using Pythagorean theorem applied to two triangles, it eliminates the height and derives an expression for the height solely in terms of the triangle's three sides, resulting in Heron's formula for the area.
The unit circle relates real numbers to points on a circle of radius 1 centered at the origin. Each real number t corresponds to a point (x, y) on the circle, allowing the definition of the six trigonometric functions in terms of x and y. The trigonometric functions sine and cosine are periodic with period 2π, meaning their values repeat every 2π units. Their domains are all real numbers and their ranges are between -1 and 1.
IJCER (www.ijceronline.com) International Journal of computational Engineeri...ijceronline
1. The document introduces the concept of a "Total Prime Graph", which is a graph that admits a special type of labeling called a "Total Prime Labeling".
2. Some properties of Total Prime Labelings are studied, and it is proved that paths, stars, bistars, combs, even cycles, helm graphs, and certain wheel graphs are Total Prime Graphs. However, odd cycles are proved to not be Total Prime Graphs.
3. The labeling must satisfy two conditions - the labels of adjacent vertices and incident edges of high degree vertices must be relatively prime. Several examples and theorems demonstrating Total Prime Graphs are provided.
The document discusses various topics in functions including:
- Types of relations such as one-to-one, one-to-many, and many-to-one.
- Ordered pairs, domains, codomains, and defining functions.
- Finding inverses of functions and identifying even and odd functions.
- Exponential and logarithmic functions including their properties and rules.
- Trigonometric and hyperbolic functions.
This document provides information on differentiation including:
- The definition of the derivative as a limit.
- Rules of differentiation including constant multiples, sums, products, quotients, and chains.
- Derivatives of trigonometric, exponential, and logarithmic functions.
- Examples of calculating derivatives using the definition and rules of differentiation.
This document provides an overview of the Laplace transform and its properties and applications. Specifically, it defines the Laplace transform and inverse Laplace transform, lists several Laplace transform pairs, and provides examples of using the Laplace transform to solve initial value problems involving differential equations with constant coefficients. It also includes a problem set with exercises calculating Laplace transforms and using them to solve initial value problems.
This document discusses partial derivatives, which are used to describe the rate of change of functions with multiple variables. It defines:
1) Partial derivatives as the rate of change of the dependent variable with respect to one independent variable, while holding other variables constant.
2) Functions of two variables have level curves where the function value is constant. Their graphs are surfaces in 3D space.
3) Higher order partial derivatives describe the rate of change of the first partial derivatives.
4) The chain rule extends differentiation to composite functions, allowing functions of variables that are themselves functions of other variables.
This document provides information on calculating limits using limit laws and discusses one-sided limits and limits at infinity. It includes theorems on limit laws and examples of applying the laws to calculate limits. There are also 36 practice problems with answers provided to find specific limits algebraically or using limit laws for rational functions, functions with noninteger or negative powers, and limits approaching positive or negative infinity.
The document discusses various topics related to vectors including:
- Definitions of vectors, scalars, magnitude and direction
- Equality of vectors and types of vectors
- Addition and subtraction of vectors using triangle law and parallelogram law
- Multiplication of a vector by a scalar
- Scalar (dot) product and properties
- Vector (cross) product and properties
- Applications to work done, moments and areas
The document provides explanations, properties, examples and formulas for key vector algebra concepts.
The document discusses various topics related to vectors including:
- Definitions of vectors, scalars, magnitude and direction
- Equality of vectors and types of vectors
- Addition and subtraction of vectors using triangle law and parallelogram law
- Multiplication of a vector by a scalar
- Scalar (dot) product and properties
- Vector (cross) product and properties
- Applications to work done by forces, area of triangles and moments
The document provides a comprehensive overview of key concepts and formulas regarding vectors and their operations.
This document defines key terms and concepts related to vectors, including:
- A vector is a quantity that has both magnitude and direction, represented by a directed line segment.
- The position vector of a point P(a,b,c) with respect to the origin (0,0,0) is denoted as OP=ai+bj+ck.
- The sum of two vectors a and b represented by the sides of a triangle taken in order is equal to the third side of the triangle taken in the opposite order, according to the triangle law of addition.
- The scalar (dot) product and cross product of vectors are defined, and properties such as commutativity and relationships to angles between vectors
This document defines key terms and concepts related to vectors, including:
- A vector is a quantity that has both magnitude and direction, represented by a directed line segment.
- The position vector of a point P(a,b,c) with respect to the origin (0,0,0) is represented as OP = ai + bj + ck.
- The sum of two vectors a and b is represented geometrically by the third side of a triangle formed by the two vectors in order.
- Scalar (dot) product and cross product are defined for two vectors, with properties such as commutativity and relationships to angles between the vectors discussed.
- Scalar triple product represents the volume of
Solution Strategies for Equations that Arise in Geometric (Clifford) AlgebraJames Smith
Drawing mainly upon exercises from Hestenes's New Foundations for Classical Mechanics, this document presents, explains, and discusses common solution strategies. Included are a list of formulas and a guide to nomenclature.
See also:
http://www.slideshare.net/JamesSmith245/rotations-of-vectors-via-geometric-algebra-explanation-and-usage-in-solving-classic-geometric-construction-problems-version-of-11-february-2016 ;
http://www.slideshare.net/JamesSmith245/resoluciones-de-problemas-de-construccin-geomtricos-por-medio-de-la-geometra-clsica-y-el-lgebra-geomtrica-vectorial ;
http://www.slideshare.net/JamesSmith245/solution-of-the-special-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/solution-of-the-ccp-case-of-the-problem-of-apollonius-via-geometric-clifford-algebra ;
http://www.slideshare.net/JamesSmith245/a-very-brief-introduction-to-reflections-in-2d-geometric-algebra-and-their-use-in-solving-construction-problems ;
http://www.slideshare.net/JamesSmith245/solution-of-the-llp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-using-reflections-and-rotations ;
http://www.slideshare.net/JamesSmith245/simplied-solutions-of-the-clp-and-ccp-limiting-cases-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/additional-solutions-of-the-limiting-case-clp-of-the-problem-of-apollonius-via-vector-rotations-using-geometric-algebra ;
http://www.slideshare.net/JamesSmith245/an-additional-brief-solution-of-the-cpp-limiting-case-of-the-problem-of-apollonius-via-geometric-algebra-ga .
This document provides an introduction to vector functions of one variable. It defines key concepts like scalar and vector, direction cosines, scalar and vector products, and differentiation of vector functions. Examples are given on determining direction cosines of a vector, the angle between vectors, and properties of triple products. The document also discusses how to determine if vectors are coplanar and visualization of differentiation of a vector function with respect to a variable like time.
Concept of Particles and Free Body Diagram
Why FBD diagrams are used during the analysis?
It enables us to check the body for equilibrium.
By considering the FBD, we can clearly define the exact system of forces which we must use in the investigation of any constrained body.
It helps to identify the forces and ensures the correct use of equation of equilibrium.
Note:
Reactions on two contacting bodies are equal and opposite on account of Newton's III Law.
The type of reactions produced depends on the nature of contact between the bodies as well as that of the surfaces.
Sometimes it is necessary to consider internal free bodies such that the contacting surfaces lie within the given body. Such a free body needs to be analyzed when the body is deformable.
Physical Meaning of Equilibrium and its essence in Structural Application
The state of rest (in appropriate inertial frame) of a system particles and/or rigid bodies is called equilibrium.
A particle is said to be in equilibrium if it is in rest. A rigid body is said to be in equilibrium if the constituent particles contained on it are in equilibrium.
The rigid body in equilibrium means the body is stable.
Equilibrium means net force and net moment acting on the body is zero.
Essence in Structural Engineering
To find the unknown parameters such as reaction forces and moments induced by the body.
In Structural Engineering, the major problem is to identify the external reactions, internal forces and stresses on the body which are produced during the loading. For the identification of such parameters, we should assume a body in equilibrium. This assumption provides the necessary equations to determine the unknown parameters.
For the equilibrium body, the number of unknown parameters must be equal to number of available parameters provided by static equilibrium condition.
This document discusses various concepts related to vectors and 3D geometry including dot products, cross products, planes, lines, and their relationships. Dot products can be used to find the angle between vectors and determine if vectors are perpendicular. Cross products give a vector perpendicular to both input vectors. Plane equations can be defined using a point and normal vector, three points, or two vectors in the plane. Lines are defined by two points or a point and direction vector. The intersection of planes and lines, parallelism, and distances between lines and points and planes are also covered.
1) Polynomial equations have as many roots as the highest power of the variable. The roots can be repeated or complex.
2) Quadratic equations can be solved by setting the coefficients equal to functions of the roots, or by factorizing the equation in terms of the roots.
3) Symmetrical functions of the roots remain the same if the roots are swapped, and can be written in terms of the sum and product of the roots.
1) Polynomial equations have as many roots as the highest power of the variable. The roots can be real or complex, and may be repeated.
2) Quadratic equations can be solved by setting the coefficients equal to functions of the roots, or by factorizing the quadratic expression.
3) Cubic equations have three roots that relate to the coefficients, and their symmetrical functions can be written in terms of sums and products of the roots.
The document provides notes on vectors, matrices, and coordinate transformations. It defines vectors and vector operations such as addition, subtraction, scalar multiplication, dot product, and cross product. It explains how to express vectors and vector operations using component form in a given coordinate system. It also discusses differentiating vectors with respect to time and how vector differentiation follows similar rules to scalar differentiation.
The document discusses dot and cross products of vectors. The dot product of two vectors A and B is defined as ABcosθ, where θ is the angle between them. It results in a scalar quantity and obeys the commutative law. The cross product of two vectors A and B is defined as ABsinθ with a direction perpendicular to A and B. It results in a vector quantity and does not obey the commutative law. The dot product is used to find projections and the angle between vectors, while the cross product is used to find the area of a parallelogram or triangle formed by vectors and the torque on a vector.
This document provides an overview of vectors and their applications in physics. It defines vectors and differentiates them from scalars, discusses vector notation and representation, and covers key concepts like addition, subtraction, and multiplication of vectors. Examples are given of vector quantities like displacement, velocity and force. The document also explains vector operators like gradient, divergence and curl, which allow converting between scalar and vector quantities, and outlines how calculus is important in physics for studying change.
This document discusses the dot product of vectors. It defines the dot product as the sum of the products of the corresponding components of two vectors. The dot product is a scalar quantity that can be used to determine the angle between vectors and whether vectors are orthogonal. It also discusses the relationship between the dot product and the projections of one vector onto another vector.
1. The document discusses three geometry problems involving vectors and their solutions:
2. It shows that the midpoint lines of a parallelogram trisect its diagonal lines.
3. It proves that if two pairs of opposite edges in a tetrahedron are perpendicular, then the third pair is also perpendicular.
4. It demonstrates that the midpoint line between two sides of a triangle is parallel to the third side and half its length.
- Points, LInes and Planes,
- Segment and Angle Addition
- Segment and Angle Bisectors
- Distance and Midpoint Formuals
- Special Angle Relationshsips
- Area
A vector has magnitude and direction. There is an algebra and geometry of vectors which makes addition, subtraction, and scaling well-defined.
The scalar or dot product of vectors measures the angle between them, in a way. It's useful to show if two vectors are perpendicular or parallel.
1. The document discusses vectors, including their magnitude and direction. Vectors can represent things like wind speed and direction, forces, velocities, etc. Magnitude represents size while direction specifies the orientation.
2. It introduces vector notation and operations. The magnitude of a vector a is written as |a|. Two vectors are equal if their magnitudes and directions are equal. Vector addition is demonstrated geometrically by drawing the vectors head to tail.
3. Examples show how to find the resultant of adding or subtracting vectors using diagrams. Vector subtraction is performed by adding the negative of the vector. Properties like the parallelogram law for vector addition are also covered.
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.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
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it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
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Introduction to AI for Nonprofits with Tapp NetworkTechSoup
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Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
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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.
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.
Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
Chapter 3( 3 d space vectors)
1. BMM 104: ENGINEERING MATHEMATICS I Page 1 of 22
CHAPTER 3: 3-D SPACE VECTORS
Basic Concept of Vector
A vector is a quantity that having a magnitude/length (absolute) and a direction.
→ →
Notation: i. AB or a is vector
→ →
ii. AB or a is modulus/absolute value/length of the vector
→ →
BA is opposite to AB
→ →
BA = − AB
→ →
Meanwhile AB = BA .
Characteristics of vectors
2. BMM 104: ENGINEERING MATHEMATICS I Page 2 of 22
→ → → → → →
1. If AB is parallel to c , then AB = k c OR c = t AB where k and t are the
scalars or parameters.
→ →
2. If k < 0 OR t < 0 , then AB and c are in the opposite directions.
3. If k > 0 OR t > 0 , then they are in the same direction.
Example:
→ →
→ 1 →
i) AB = 2 c or c= AB
2
→
ii) 1→ or
→ →
AB = − d d = −2 AB
2
Addition Law of Vectors
→ → → →
Let c = AB , d = BC . Refer to below diagram.
→ → → → →
AC = AB + BC = c + d
→ →
If OA is the position vector for A and OB is the position vector for B then
→ → →
OA+ AB = OB
→ → → → → →
AB = OB − OA BUT AB ≠ OA −OB
→
Example: Find NS = ?
Components of Vectors
2-D Space
3. BMM 104: ENGINEERING MATHEMATICS I Page 3 of 22
i j
2 basic vectors : ~ and ~
They are also called unit vectors as i =1 and j =1
~ ~
→
Position vector of point A( a ,b ) is given by OA = a ~ + b j .
i
~
→
Absolute/ modulus OA = a 2 + 2
b by Pythagoras Theorem
Example:
→
i) OA = −3 i + 4 j
~ ~
→
OA = 9 +16 =5
3-D Space
4. BMM 104: ENGINEERING MATHEMATICS I Page 4 of 22
o All the x-axes, y-axes are perpendicular to each other.
o There are 3 basic vectors: i , j , k .
~ ~ ~
o They are all unit vectors that parallel to the axes respectively and thus they also
perpendicular to each other.
a
→
Position vector of A( a ,b , c ) is OA = a ~ + b j + c k = ( a ,b , c ) = b and
i
~
~
c
→
Length is OA = a 2 + 2 + 2
b c
3
→ →
Example: Given OA = 4 . Find OA =?
6
Addition of vectors for 3-D Space
→ →
The sum of two vector OA = ( a1 , a 2 , a 3 ) and OB = ( b1 ,b2 ,b3 ) is the vector formed by
adding the respective component;
5. BMM 104: ENGINEERING MATHEMATICS I Page 5 of 22
→ →
OA+ OB = ( a1 + b1 , a 2 + b2 , a 3 + b3 )
Subtraction of vectors for 3-D Space
→ →
The subtraction of two vectors OA = ( a1 , a 2 , a 3 ) and OB = ( b1 ,b2 ,b3 ) is the vector
formed by adding the respective component;
→ → →
AB = OB − OA
→ →
OB − OA = ( b1 − a1 ,b2 − a 2 ,b3 − a 3 )
Example: Given A( −1,1,4 ) , B ( 8 ,0 ,2 ) and C ( 5 ,− ,11) . Find
2
→
(i) OA
→
(ii) OB
→
(iii) OC
→
(iv) AB
→
(v) AC
→
(vi) AB
→
(v) AC
→
Unit Vector v~
in the Direction of v
→
→
→
v
v=
~
→
→
is a unit vector in the direction of v .
v
6. BMM 104: ENGINEERING MATHEMATICS I Page 6 of 22
− 1
→ →
Example: Given v = 2 . Find unit vector in the direction of v ?
3
Dot/ Scalar Product
→ →
Notation for Dot product: a • b ∈ ℜ
→ →
The dot product of a = ( a1 , a 2 , a 3 ) and b = ( b1 ,b2 ,b3 ) is the real number a • b obtained
→ →
by
→ → →→
a• = a
b b cos θ
where θ is the angle between a and b and 0 ≤ θ ≤ π .
→ →
When we measuring angle between two vectors, the vectors must have the same initial
point.
→ → → →
Example: (i) Given a =6 and b =7. Find a • b .
→ →
= and θ =
π → →
(ii) Given a =5 and b 4 . Find a • b .
4
7. BMM 104: ENGINEERING MATHEMATICS I Page 7 of 22
Example:
i • i = i i cos 0
~ ~ ~ ~
i • i = 1 ×1 ×1 = 1
~ ~
Important:
(a) j• j = k • k = i • i = 1
~ ~ ~ ~ ~ ~
(b) i• j = i
~ ~ ~
j cos 90
~
(c) i• j = 1×1×0 = 0
~ ~
(d) i • j = 0 = i • k = j• k
~ ~ ~ ~ ~ ~
.
→ →
→ →
If a ⊥ b then a • b = a b cos 90 = 0 .
~ ~
Note:
8. BMM 104: ENGINEERING MATHEMATICS I Page 8 of 22
~ ~ → →
(i) If a• b = 0 then a ⊥ b
→ → → →
(ii) a× b ≠ b× a
Properties of the dot product
→ → → →
a) a• b = b• a : Cumulative
→ → → → → →
b •a = b a cos θ=a •b
→
→ → → → → →
b) a •b + c = a • b + a • c
: Distributive
→ → → → → →
c) k a • b = k a • b = a •k b
Example:
5 8
→
→ → →
Finding angle between the vectors a = 3 and b = − 9 given that a • b = −9 .
− 2 11
Formula to compute scalar product
a1 b1
→ →
a • b = a 2 • b2
a b
3 3
→ →
a • b = a1b1 + a 2 b2 + a 3 b3
3 − 2
→→ → → → →
Example: Given a = 4 and b = 6 . Verify a • b = b • a .
− 1 3
Remark: By using addition law of vectors and the law of cosine, it can be showed that
→ →
a b + b + b = a b cos θ.
a 1 a
1 2 2 3 3
→
→ →
Component of a in the direction of n : Denoted as comp n a →
9. BMM 104: ENGINEERING MATHEMATICS I Page 9 of 22
OQ
cos θ =
OP
OQ = OP cos θ
a cos θ
→
=
→ →
OQ is called component of vector a in the direction of n
→
OQ = Comp→ a
n
= OP cos θ
= a cos θ
→
^
n=1
→ ^
= a cos θ ⋅ n since
~
~
By the definition of product
→ → ^
comp → a = a • n
n ~
Example:
Find
2 1
→ →
→
(i) comp → a given a = 1 and n = 1 .
n
1 1
10. BMM 104: ENGINEERING MATHEMATICS I Page 10 of 22
2 1
→ → →
(ii) comp → a given a = − 1 and n = 1 .
n
−7 1
Application of Dot Product : WORK DONE
Work done = Magnitude of force in the direction of motion times the distance it travels
= F cos θPQ
→ →
→ →
W =F •PQ
1
→
Example: A force F = − 2 causes a body to move from P (1,− ,2 ) to Q (7 ,3 ,6 ) .
1
3
Find the work done by the force.
Vector/ Cross Product
→ →
Definition: a×b is defined as a vector that
→ → → →
1. a×b is perpendicular to both a and b .
→ → → → → →
( a× b ) • a = 0 and ( a× b ) • b = 0
11. BMM 104: ENGINEERING MATHEMATICS I Page 11 of 22
→ → → →
2. Direction of a×b follows right-handed screw turned from a to b
→ → → →
b× a = − a× b
b sin θ
→→
→ →
3. Modulus of a×b is a
Remarks:
a sin θ
→→
→ →
Modulus of b×a is therefore b
→ → → →
a× b ≠ b× a
→ → → → → → → →
a× b = − b × a or b× a = − a× b
12. BMM 104: ENGINEERING MATHEMATICS I Page 12 of 22
→ →
a×b ∧
=e
Unit Vector : → →
a×b
~
→ → → → ∧ → → ∧
a×b = a×b e = a b sin θ e
~ ~
4. i× i = 0 j× j = 0 k× k = 0
~ ~ ~ ~ ~ ~
i× j = i
~ ~ ~
j sin 90 k =k
~ ~ ~
and ~ ~ ~ ~
( )
j×i = j i sin 90 −k = −k
~ ~
Similarly, j× k = i and k× j = −i
~ ~ ~ ~ ~ ~
k× i = j and i× k = − j
~ ~ ~ ~ ~ ~
→ →
Determinant Formula for a×b
13. BMM 104: ENGINEERING MATHEMATICS I Page 13 of 22
i j k
→ → ~ ~ ~
a× b = a1 a2 a3
b1 b2 b3
a2 a3 a1 a3 a1 a2
= i− j+ k
b2 a3 ~ b1 b3 ~ b1 b2 ~
= ( a 2 b3 − a 3 b2 ) i − ( a1b3 − a 3 b1 ) ~ + ( a1b2 − a 2 b1 ) k
~
j
~
3
→ → → → →
→
→ →
→
Example: Find a×b , b×a and verify that a× b = −b× a given a = − 4 and
2
9
→
b = −6 .
2
Applications of Cross Product
1. The moment of a force
→ →
A force F is applied at a point with position vector r to an object causing the object to
rotate around a fixed axis.
14. BMM 104: ENGINEERING MATHEMATICS I Page 14 of 22
As the magnitude of moment of the force at O is
M 0 = F • d = ( Magnitude of force perpendicular to d ) × (Magnitude of displacement)
→ → → →
Thus we have M 0 = F sin θ• r = r×F
Therefore we define the moment of the force about O as the vector
→ → →
M 0 = r×F
→ → → → →
As r×F =r F sin θ= M 0 =M 0
→
The magnitude, M0 , is a measure of the turning effect of the force in unit of Nm.
2
→
Example: Calculate the moment about O of the force F = 3 that is applied at the point
1
with position vector 3j. Then calculate its magnitude.
2. Calculate the area of a triangle
15. BMM 104: ENGINEERING MATHEMATICS I Page 15 of 22
By the sine rule:
1 1 → →
Area of ∆ABC = 2 bc sin A = 2 AC× AB
1 1 → →
= 2 ac sin B = 2 BC× BA
1 1 → →
= 2 ab sin C = 2 CB×CA
Example: Find area for a triangle with vertices A(0 ,7 ,1) , B (1,3 ,2 ) and C ( − 2 ,0 ,3 ) .
Equations (Vector, parametric and Cartesian equations) of a line
16. BMM 104: ENGINEERING MATHEMATICS I Page 16 of 22
x v1
→ → →
Let r ( t ) = OP = y and v = v 2 as a vector that parallel to the line L.
z v
3
→ → → →
As P0 P // v thus P0 P = t v , t is a parameter (scalar).
By the addition law of vectors, we obtain
→ → →
OP = OP 0 + P0 P
i.e. the vector equation of a line passing through a fixed point P0 and parallel to a vector
→ → → →
v is r ( t ) = OP 0 + t v .
x a v1
y = b + t v2
z c v
3
x a + tv1
y = b + tv 2
z c + tv
3
⇒ x = a + tv1 , y = b + tv 2 , z = c + tv 3 are the parametric equations of L.
x −a y −b z −c
⇒ = = is the Cartesian equation of L.
v1 v2 v3
17. BMM 104: ENGINEERING MATHEMATICS I Page 17 of 22
Example: Find the vector equation of line passes through A( 3,2 ) and B(7 ,5 ) .
Example: Find Cartesian equation of line passes through A( 5 ,− ,3 ) and B ( 2 ,1,− ) .
2 4
Equation (Vector and Cartesian equations) of a plane
n1 x n1 a
Vector equation: n2 • y = n2 • b
n z n c
3 3
Cartesian equation: n1 x + n 2 y + n 3 z = n1 a + n 2 b + n 3 c
Remark: In general, Ax + By + Cz = D OR Ax + By + Cz = 1 is the Cartesian equation of a
A
plane with a normal vector B .
C
18. BMM 104: ENGINEERING MATHEMATICS I Page 18 of 22
Example: A plane contains A(1,0 ,1) , B ( − 2 ,5 ,0 ) and C ( 3 ,1,1) . Find the vector and
Cartesian forms of the equation of the plane.
→ →
AB = −3 i + 5 j − k and AC = 2 i + j
~ ~ ~ ~ ~
Distance From A Point to A Line and to A Plane
Distance From A Point to A Line
→ → →
Distance, d, from a fixed point P to a line: r ( t ) = OA+ t v where A is a point on the line
→
and v is a vector parallel to the line is given by
→ → ∧ → ∧
d = AP sin θ = AP v sin θ = AP×v
~ ~
1−t
→
Example: Find the distance from point ( 4 ,3 ,2 ) to the line L : r ( t ) = 2 + 3t .
3 +t
Distance From A Point to A Plane
19. BMM 104: ENGINEERING MATHEMATICS I Page 19 of 22
→ → → →
Distance, D, from a fixed point P to a plane n • r = n • OA where A is a point on the plane
→
and n is a normal vector to the plane is given by
→
→ → ∧ ∧
D = AP cos θ = AP n cos θ = AP •n .
~ ~
Example: Find the distance from point ( 4 ,− ,3 ) to the plane x + 3 y − 6 z = 9 .
3
PROBLEM SET: 3-D SPACE VECTORS
1. Points P , Q and R have coordinates ( 9 ,1,0 ) , ( 8 ,− ,5 ) and C ( 5 ,5 ,7 )
3
respectively. Find
(a) the position vectors of P, Q and R.
→ →
(b) PQ and QR .
20. BMM 104: ENGINEERING MATHEMATICS I Page 20 of 22
→ →
(c) PQ and QR .
2. A triangle has vertices A(1,3 ,2 ) , B ( −1,5 ,9 ) and C ( 2 ,7 ,1) respectively.
Calculate the vectors which represent the sides of the triangle.
→ → → → → →
3. Find a • b and verify that a • b = b • a if
2 3 1 3
→ → → →
(a) a = − 5 , b = 2 (b) a = 8 , b = − 2
0 0 7 5
4. (a) Find the component of the vector 2 ~ + ~ + 7 k in the direction of the vector
i j
~
i + j+ k .
~ ~ ~
(b) Find the component of the vector 7 ~ + 2~ j − k in the direction of the vector
i
~
i − j+ 2 k .
~ ~ ~
→
5. A force F = 3 i + 7 k causes a body to move from point A(1,1,2 ) to point
~ ~
B (7 ,3 ,5 ) .
Find the work done by the force.
3 5
→ → → →
6. (a) If a = 2 and b = − 1 , find a×b .
− 1 − 1
→ → → →
(b) Verify that a× b = −b× a .
7. (a) Find the area of the triangle with vertices P (1,2 ,3 ) , Q( 4 ,− ,2 ) and
3
R ( 8 ,1,1) .
→
(b) A force F of magnitude 2 units acts in the same direction of the vector
3 i − 2 j + 4 k . It causes a body to move from point S ( −2 ,− ,− ) to
3 4
~ ~ ~
point T (7 ,6 ,5 ) . Find the work done by the force.
8. Find the vector equation of the line passing through
(a) A( 3 ,2 , ) and B ( −1,2 ,3 ) .
7
→ →
(b) the points with position vectors p = 3 i + 7 k − 2 k and q = −3 ~ + 2 j + 2 k .
i
~
~ ~ ~ ~
Find also the cartesian equation of this line.
21. BMM 104: ENGINEERING MATHEMATICS I Page 21 of 22
1
(c) ( 9 ,1,2 ) and which is parallel to the vector 1 .
1
9. Given A( 9 ,1,1) , B ( 8 ,1,1) and C ( 9 ,0 ,2 ) . Find
(a) the area of the triangle ABC.
(b) the Cartesian equation of the plane containing A, B and C.
4 −t
→
10. (a) Find the distance from point (6 ,3 ,5 ) to the line L : r ( t ) = − 3 + 2t .
2 + 5t
(b) Find the distance from point ( 4 ,− ,6 ) to the plane 2 x − 4 y + 3 z = 8 .
3
−7
→
11. (a) Find the distance from A( 2 ,1,− ) to the plane 5 • r ( t ) = 10.
3
6
y +4 z −3
(b) Find the distance from point B (6 ,3 ,− ) to the line L :
5 = ,x = 5
3 2
.
ANSWERS FOR PROBLEM SET: 3-D SPACE VECTORS
9 8 5
→ → →
1. (a) OP = 1 , OQ = − 3 , OR = 5
0 5 7
−1 − 3
(b) − 4 , 8 (c) 42 , 77
5 2
− 2 3 1
→ → →
2. AB = 2 , BC = 2 , AC = 4
7 − 8 − 1