Calculus II - 33
•Download as KEY, PDF•
2 likes•392 views
The dot product of two vectors a and b is defined as a·b and is calculated by multiplying their corresponding components and adding them. It can be used to calculate the projection of one vector onto another and the cosine of the angle between the vectors. The cross product of two vectors a and b is defined as axb and produces a third vector that is perpendicular to both a and b. It has various properties including being anticommutative and having a magnitude equal to the area of the parallelogram determined by the two vectors.
Report
Share
Report
Share
Recommended
Calculus II - 34
The cross product and dot product of vectors are introduced. Properties of the cross product include that it is only defined for 3D vectors and that the magnitude of the cross product equals the area of the parallelogram determined by the two vectors. Vector, parametric, and various forms of plane equations are presented along with examples of finding equations of lines and planes given conditions.
Calculus II - 32
A vector is a quantity that has both magnitude and direction. It can be represented in a coordinate system with points (x,y,z) and the vector between two points is defined as (x2-x1, y2-y1, z2-z1). Vectors can be added, subtracted, and multiplied by scalars. The length of a vector is calculated using the Pythagorean theorem and vectors can be unitized. There are standard basis vectors i, j, k in 3D space and vectors can be represented as a combination of these basis vectors. The dot and cross products can be used to calculate properties of vectors.
Calculus II - 35
The document discusses representations of lines, planes, and space curves in vector calculus. It defines the vector equation of a line, parametric equation of a line, and symmetric equation of a line. It also defines the vector equation and scalar equation of a plane. It then discusses representing space curves using parametric vector functions and calculating derivatives and integrals of vector functions, including rules for differentiation. It provides an example problem proving that the tangent line to a circle at a point is perpendicular to the line connecting that point to the circle's center.
CST 504 Distance in the Cartesian Plane
This document discusses different methods for calculating distance in the Cartesian plane. It introduces horizontal, vertical, and oblique (diagonal) distance. Horizontal distance is found by subtracting the x-coordinates of two points, vertical by subtracting the y-coordinates, and oblique distance uses the Pythagorean theorem. The distance formula is also provided as a method to find the distance between any two points by taking the square root of the sum of the differences of the x-coordinates and y-coordinates squared. The document also covers finding the midpoint between two points by taking the average of the x-coordinates and y-coordinates.
VECTOR ANALYSIS- 2
1. The document discusses various vector algebra concepts including vector projection, work done by a force, compound angle formulas, and vector cross and dot products.
2. Vector projection and work done formulas are derived. The compound angle formulas for cosine and sine are proved using vector diagrams.
3. Cross and dot products are explained. Various applications are covered including proving trigonometric identities like the sine rule and finding areas and volumes.
8 arc length and area of surfaces x
The document discusses calculating the length of a curve defined by a function f(x). It explains that the length can be approximated by partitioning the domain into small intervals, drawing line segments between the corresponding points on the curve, and taking the sum of the lengths. This sum approaches the true curve length as the interval size approaches 0. Using the Mean Value Theorem, the sum can be written as an integral involving the derivative of f, known as the arc length formula. While difficult generally, the integral can be solved for some basic functions like polynomials.
Circle generation algorithm
The document describes the Breshenham's circle generation algorithm. It explains that the algorithm uses a decision parameter to iteratively select pixels along the circumference of a circle. It provides pseudocode for the algorithm, which initializes x and y values, calculates a decision parameter, and increments x while decrementing y at each step, plotting points based on the decision parameter. An example of applying the algorithm to generate a circle with radius 5 is also provided.
11 equations of planes
The document discusses equations of planes in 3D space. It introduces the point-normal form of a plane equation: A(x-r) + B(y-s) + C(z-t) = 0, where <A,B,C> is a normal vector to the plane and <r,s,t> is a point on the plane. This equation is analogous to the point-slope form for a line, and can be derived by setting the dot product of the normal vector and a position vector from the point <r,s,t> to a generic point <x,y,z> on the plane equal to zero.
Recommended
Calculus II - 34
The cross product and dot product of vectors are introduced. Properties of the cross product include that it is only defined for 3D vectors and that the magnitude of the cross product equals the area of the parallelogram determined by the two vectors. Vector, parametric, and various forms of plane equations are presented along with examples of finding equations of lines and planes given conditions.
Calculus II - 32
A vector is a quantity that has both magnitude and direction. It can be represented in a coordinate system with points (x,y,z) and the vector between two points is defined as (x2-x1, y2-y1, z2-z1). Vectors can be added, subtracted, and multiplied by scalars. The length of a vector is calculated using the Pythagorean theorem and vectors can be unitized. There are standard basis vectors i, j, k in 3D space and vectors can be represented as a combination of these basis vectors. The dot and cross products can be used to calculate properties of vectors.
Calculus II - 35
The document discusses representations of lines, planes, and space curves in vector calculus. It defines the vector equation of a line, parametric equation of a line, and symmetric equation of a line. It also defines the vector equation and scalar equation of a plane. It then discusses representing space curves using parametric vector functions and calculating derivatives and integrals of vector functions, including rules for differentiation. It provides an example problem proving that the tangent line to a circle at a point is perpendicular to the line connecting that point to the circle's center.
CST 504 Distance in the Cartesian Plane
This document discusses different methods for calculating distance in the Cartesian plane. It introduces horizontal, vertical, and oblique (diagonal) distance. Horizontal distance is found by subtracting the x-coordinates of two points, vertical by subtracting the y-coordinates, and oblique distance uses the Pythagorean theorem. The distance formula is also provided as a method to find the distance between any two points by taking the square root of the sum of the differences of the x-coordinates and y-coordinates squared. The document also covers finding the midpoint between two points by taking the average of the x-coordinates and y-coordinates.
VECTOR ANALYSIS- 2
1. The document discusses various vector algebra concepts including vector projection, work done by a force, compound angle formulas, and vector cross and dot products.
2. Vector projection and work done formulas are derived. The compound angle formulas for cosine and sine are proved using vector diagrams.
3. Cross and dot products are explained. Various applications are covered including proving trigonometric identities like the sine rule and finding areas and volumes.
8 arc length and area of surfaces x
The document discusses calculating the length of a curve defined by a function f(x). It explains that the length can be approximated by partitioning the domain into small intervals, drawing line segments between the corresponding points on the curve, and taking the sum of the lengths. This sum approaches the true curve length as the interval size approaches 0. Using the Mean Value Theorem, the sum can be written as an integral involving the derivative of f, known as the arc length formula. While difficult generally, the integral can be solved for some basic functions like polynomials.
Circle generation algorithm
The document describes the Breshenham's circle generation algorithm. It explains that the algorithm uses a decision parameter to iteratively select pixels along the circumference of a circle. It provides pseudocode for the algorithm, which initializes x and y values, calculates a decision parameter, and increments x while decrementing y at each step, plotting points based on the decision parameter. An example of applying the algorithm to generate a circle with radius 5 is also provided.
11 equations of planes
The document discusses equations of planes in 3D space. It introduces the point-normal form of a plane equation: A(x-r) + B(y-s) + C(z-t) = 0, where <A,B,C> is a normal vector to the plane and <r,s,t> is a point on the plane. This equation is analogous to the point-slope form for a line, and can be derived by setting the dot product of the normal vector and a position vector from the point <r,s,t> to a generic point <x,y,z> on the plane equal to zero.
9 determinant and cross product
The document discusses matrices and determinants. It defines a matrix as a rectangular table with numbers or formulas as entries. It provides examples of 2x2 and 3x3 matrices. The document explains that square matrices have a number called the determinant extracted from them. It then discusses how the 2x2 determinant represents the signed area of a parallelogram defined by the row vectors, and explores properties of the sign of the determinant. Finally, it suggests generalizing these concepts to 3x3 determinants.
Mathnumberplane
The document discusses straight line graphs on the number plane and their equations.
1. The red lines are parallel with equations that have -1/2, which determines the angle. The sign of the constant a indicates whether the line increases or decreases as x increases.
2. The blue lines are also parallel, with equations sharing the constant 2, which determines their angle.
3. The lines y=-1/2x, y=2x, and y=x intersect at the origin because their equations have b=0, making the y-value 0 when x=0.
The document then provides two alternative ways to graph a line: i) finding the x- and y-
Notes 3-2
The document discusses graph translations. It defines a translation as moving each point (x,y) on a graph to a new location (x+h, y+k) using horizontal and vertical magnitudes h and k. Examples show finding the rule for a translation based on a point mapping, translating points between graphs, and rewriting equations in vertex form to identify translations between parabolas. The key aspects are that translations shift graphs in the xy-plane using h and k values and the graph-translation theorem allows identifying translations by replacing x with x-h and y with y-k in equations.
MT T4 (Bab 3: Fungsi Kuadratik)
This document discusses quadratic functions and their graphs. It begins by defining the general form of a quadratic function as f(x) = ax2 + bx + c, where a ≠ 0. It then explains how to identify the shape of a quadratic graph based on the sign of a, whether it is positive or negative. Examples are provided to show how to sketch graphs, find maximum and minimum values, axes of symmetry, and zeros. The document also covers using the discriminant to determine the number and type of roots, and completing the square to find the vertex of a quadratic function.
Solucionario teoria-electromagnetica-hayt-2001
This document provides solutions to problems from Chapter 1 of an electromagnetism textbook. It solves for vectors, vector operations, vector fields and their properties at different points. Some key points solved include finding unit vectors, magnitudes of vectors, angles between vectors, vector projections, and visualizing vector fields. The document contains step-by-step workings for 12 multi-part problems involving concepts of vector algebra and vector calculus.
Sec 3 E Maths Notes Coordinate Geometry
This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.
The Digital Image Processing Q@A
This document contains questions and answers about image processing concepts such as linear indexing, converting between m-paths and 4-paths, adjacency in image subsets, shortest path lengths between pixels using different adjacency types, and inverse affine transformations including scaling, translation, shearing, and rotation. Equations and examples are provided to derive the inverse transformations from the original transformations.
1422798749.2779lecture 5
The document discusses various methods for 3D object modeling and representation, including:
- Polygonal meshes which approximate surfaces and solids using polygons and can represent a broad class of objects.
- Solid modeling using polygonal meshes where directional information is added to faces using normal vectors.
- Sweep representations that form shapes by extruding or sweeping 2D profiles through space.
- Surface modeling using explicit functions of two variables or surfaces of revolution obtained by rotating curves around axes.
Graph functions
The document provides examples and explanations for sketching the graphs of various types of functions, including:
1) Linear functions, which produce straight lines. The slope and y-intercept determine the graph.
2) Quadratic functions, which produce parabolas. The direction of opening and intercepts are used to sketch the graph.
3) Cubic functions, which produce S-shaped curves. The direction of turning and intercepts are considered.
4) Reciprocal functions, which produce hyperbolas. The direction and intercepts are the key factors for the graph.
Step-by-step methods are outlined for accurately sketching graphs of each function type based on their defining characteristics.
07object3d
3D objects can be represented in several ways including point clouds, surfaces, and solids. Surfaces include meshes composed of polygons, subdivision surfaces defined by rules, and parametric or implicit surfaces with mathematical equations. Solids include voxel grids, binary space partition trees, constructive solid geometry using boolean operations, and swept volumes along trajectories. Each representation has advantages for different applications like rendering, modeling, or 3D scanning.
12 quadric surfaces
The document discusses the trace-method for visualizing quadric surfaces. A trace is the intersection of a surface with a plane. For a given surface S, traces parallel to the coordinate planes reveal the shape of S. Specifically, the intersection of S with planes where x=c, y=c, or z=c produce traces perpendicular to the respective axes. The traces of quadric surfaces are conic sections, such as ellipses or circles, which can be used to sketch the overall surface shape.
Digtial Image Processing Q@A
This document contains answers to multiple questions about image processing concepts. For question 22a, the kernel formed by the outer product of vectors v and wT is determined to be separable. For question 22b, it is explained that a separable kernel w can be decomposed into two simpler kernels w1 and w2 such that w = w1 * w2. This allows the convolution to be computed more efficiently in two steps by first convolving w1 with the image and then convolving the result with w2, requiring fewer operations than a direct convolution with w.
Chapter 4 Integration
Chapter 4 IntegrationSchool of Design Engineering Fashion & Technology (DEFT), University of Wales, Newport
This document provides an introduction to integration (calculus) as taught in an undergraduate engineering course. It defines integration as the reverse process of differentiation and describes how it can be used to find the area under a curve. The document outlines key integration terminology like indefinite integrals, definite integrals, and the constant of integration. It also provides examples of integrating common functions and using integration to calculate volumes of solids of revolution.Applications of Integrations
The document discusses various applications of the definite integral, including finding the area under a curve, the area between two curves, and the volume of solids of revolution. It provides examples of calculating each of these, such as finding the area between the curves y=x and y=x5 from x=-1 to x=0. It also explains how to set up definite integrals to calculate volumes when rotating an area about the x- or y-axis. In conclusion, it states that integrals can represent areas or generalized areas and are fundamental objects in calculus, along with derivatives.
Computer graphics
The document describes algorithms for 2D transformations in computer graphics using C programming language. It discusses functions for translation, rotation, and scaling of 2D objects.
The algorithms take input for the type of transformation, parameters for the transformation, and coordinates of the 2D object. Translation simply adds the translation offsets to the x and y coordinates. Rotation rotates the object around the x-axis or y-axis based on the angle input. Scaling multiplies the x and y coordinates by respective scaling factors.
The transformed object is displayed on the graphics screen. Thus basic 2D transformations like translation, rotation, scaling are implemented through these algorithms in C.
Random Number Generation
The document contains solutions to 3 problems involving generating random numbers using different linear congruential methods. The first problem uses the linear congruential method to generate 3 two-digit random integers: 63, 51, and 55. The second problem uses the multiplicative congruential method to generate 4 three-digit random integers: 333, 319, 717, and 831. The third problem uses the mixed congruential method to generate 3 two-digit random numbers: 88, 45, and 44.
IRJET- On Greatest Common Divisor and its Application for a Geometrical S...
This document discusses greatest common divisors (GCD) and their application to geometric structures. It begins by defining GCD and least common multiple (LCM) and proving several properties about them. It then derives formulas for the minimum number of identical rectangular tiles or cuboids needed to construct a square floor or cube, respectively. Specifically, it shows the minimum number of tiles/cuboids is equal to the GCD of certain parameters divided by the area/volume of each tile/cuboid. The document provides examples to illustrate the formulas.
Finite fields
Finite fields are important in cryptography. They involve arithmetic operations on a finite set of elements called a Galois field. Two important examples are GF(p), the integers modulo a prime p, and GF(2n), polynomials modulo an irreducible polynomial of degree n with coefficients in GF(2). Polynomial arithmetic in GF(2n) can be performed using bitwise XOR and shift operations, with modulo reduction using the irreducible polynomial. This allows efficient computation in finite fields important for cryptography.
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
Christian Jett Morales presents on the quadratic function derived from zeros of the function. The presentation includes:
- An introduction to deriving the quadratic function f(x) = ax^2 + bx + c from the zeros r1 and r2 of the function.
- Examples of deriving quadratic functions when given different zeros, such as 3/2 and -1, or 2+√3 and 2-√3.
- An activity for participants to practice deriving quadratic functions.
- A dedication to God and the presenter's teacher for the opportunity to create the presentation.
Calculus II - 25
This document discusses tests for determining if infinite series converge or diverge, including:
1) The Alternating Series Test, which can be used to determine if alternating series (with terms that are alternately positive and negative) converge.
2) Absolute and conditional convergence, where a series is absolutely convergent if the sum of the absolute values converges, and conditionally convergent if the series converges but the sum of absolute values does not.
3) The Ratio and Root Tests, which can be used to determine if series converge absolutely (and thus converge) based on the ratios or roots of successive terms.
Calculus II - 36
The document discusses arc length, curvature, and formulas for calculating them for curves in 2D and 3D. It provides the definitions of arc length, unit tangent vector, and curvature. Examples are given of finding the curvature of various curves, such as circles, helices, and parabolas, at given points using parametrizations of the curves. In general, the curvature of a plane curve f(x) = y is k(x) = f''(x) / [1+(f'(x))^2]^(3/2).
Calculus II - 31
The document discusses 3D coordinate systems and vectors. It defines:
1) The distance formula in 3D space between two points (x1,y1,z1) and (x2,y2,z2) as the square root of (x2-x1)2 + (y2-y1)2 + (z2-z1)2.
2) Vectors as quantities that have both magnitude and direction. It shows examples of vector addition and subtraction using the triangle and parallelogram laws.
3) Key properties of vectors including that vector addition is commutative and distributive over vector addition, scalar multiplication distributes over vector addition, and vectors
More Related Content
What's hot
9 determinant and cross product
The document discusses matrices and determinants. It defines a matrix as a rectangular table with numbers or formulas as entries. It provides examples of 2x2 and 3x3 matrices. The document explains that square matrices have a number called the determinant extracted from them. It then discusses how the 2x2 determinant represents the signed area of a parallelogram defined by the row vectors, and explores properties of the sign of the determinant. Finally, it suggests generalizing these concepts to 3x3 determinants.
Mathnumberplane
The document discusses straight line graphs on the number plane and their equations.
1. The red lines are parallel with equations that have -1/2, which determines the angle. The sign of the constant a indicates whether the line increases or decreases as x increases.
2. The blue lines are also parallel, with equations sharing the constant 2, which determines their angle.
3. The lines y=-1/2x, y=2x, and y=x intersect at the origin because their equations have b=0, making the y-value 0 when x=0.
The document then provides two alternative ways to graph a line: i) finding the x- and y-
Notes 3-2
The document discusses graph translations. It defines a translation as moving each point (x,y) on a graph to a new location (x+h, y+k) using horizontal and vertical magnitudes h and k. Examples show finding the rule for a translation based on a point mapping, translating points between graphs, and rewriting equations in vertex form to identify translations between parabolas. The key aspects are that translations shift graphs in the xy-plane using h and k values and the graph-translation theorem allows identifying translations by replacing x with x-h and y with y-k in equations.
MT T4 (Bab 3: Fungsi Kuadratik)
This document discusses quadratic functions and their graphs. It begins by defining the general form of a quadratic function as f(x) = ax2 + bx + c, where a ≠ 0. It then explains how to identify the shape of a quadratic graph based on the sign of a, whether it is positive or negative. Examples are provided to show how to sketch graphs, find maximum and minimum values, axes of symmetry, and zeros. The document also covers using the discriminant to determine the number and type of roots, and completing the square to find the vertex of a quadratic function.
Solucionario teoria-electromagnetica-hayt-2001
This document provides solutions to problems from Chapter 1 of an electromagnetism textbook. It solves for vectors, vector operations, vector fields and their properties at different points. Some key points solved include finding unit vectors, magnitudes of vectors, angles between vectors, vector projections, and visualizing vector fields. The document contains step-by-step workings for 12 multi-part problems involving concepts of vector algebra and vector calculus.
Sec 3 E Maths Notes Coordinate Geometry
This document provides examples and explanations of using the distance formula and equations of lines in coordinate geometry. It defines the distance formula and shows how to calculate the distance between two points with given coordinates. It also demonstrates how to determine the gradient and y-intercept of a line given its equation, find the equation of a line given the gradient and a point or two points, and find values related to lines parallel or intersecting given lines.
The Digital Image Processing Q@A
This document contains questions and answers about image processing concepts such as linear indexing, converting between m-paths and 4-paths, adjacency in image subsets, shortest path lengths between pixels using different adjacency types, and inverse affine transformations including scaling, translation, shearing, and rotation. Equations and examples are provided to derive the inverse transformations from the original transformations.
1422798749.2779lecture 5
The document discusses various methods for 3D object modeling and representation, including:
- Polygonal meshes which approximate surfaces and solids using polygons and can represent a broad class of objects.
- Solid modeling using polygonal meshes where directional information is added to faces using normal vectors.
- Sweep representations that form shapes by extruding or sweeping 2D profiles through space.
- Surface modeling using explicit functions of two variables or surfaces of revolution obtained by rotating curves around axes.
Graph functions
The document provides examples and explanations for sketching the graphs of various types of functions, including:
1) Linear functions, which produce straight lines. The slope and y-intercept determine the graph.
2) Quadratic functions, which produce parabolas. The direction of opening and intercepts are used to sketch the graph.
3) Cubic functions, which produce S-shaped curves. The direction of turning and intercepts are considered.
4) Reciprocal functions, which produce hyperbolas. The direction and intercepts are the key factors for the graph.
Step-by-step methods are outlined for accurately sketching graphs of each function type based on their defining characteristics.
07object3d
3D objects can be represented in several ways including point clouds, surfaces, and solids. Surfaces include meshes composed of polygons, subdivision surfaces defined by rules, and parametric or implicit surfaces with mathematical equations. Solids include voxel grids, binary space partition trees, constructive solid geometry using boolean operations, and swept volumes along trajectories. Each representation has advantages for different applications like rendering, modeling, or 3D scanning.
12 quadric surfaces
The document discusses the trace-method for visualizing quadric surfaces. A trace is the intersection of a surface with a plane. For a given surface S, traces parallel to the coordinate planes reveal the shape of S. Specifically, the intersection of S with planes where x=c, y=c, or z=c produce traces perpendicular to the respective axes. The traces of quadric surfaces are conic sections, such as ellipses or circles, which can be used to sketch the overall surface shape.
Digtial Image Processing Q@A
This document contains answers to multiple questions about image processing concepts. For question 22a, the kernel formed by the outer product of vectors v and wT is determined to be separable. For question 22b, it is explained that a separable kernel w can be decomposed into two simpler kernels w1 and w2 such that w = w1 * w2. This allows the convolution to be computed more efficiently in two steps by first convolving w1 with the image and then convolving the result with w2, requiring fewer operations than a direct convolution with w.
Chapter 4 Integration
Chapter 4 IntegrationSchool of Design Engineering Fashion & Technology (DEFT), University of Wales, Newport
This document provides an introduction to integration (calculus) as taught in an undergraduate engineering course. It defines integration as the reverse process of differentiation and describes how it can be used to find the area under a curve. The document outlines key integration terminology like indefinite integrals, definite integrals, and the constant of integration. It also provides examples of integrating common functions and using integration to calculate volumes of solids of revolution.Applications of Integrations
The document discusses various applications of the definite integral, including finding the area under a curve, the area between two curves, and the volume of solids of revolution. It provides examples of calculating each of these, such as finding the area between the curves y=x and y=x5 from x=-1 to x=0. It also explains how to set up definite integrals to calculate volumes when rotating an area about the x- or y-axis. In conclusion, it states that integrals can represent areas or generalized areas and are fundamental objects in calculus, along with derivatives.
Computer graphics
The document describes algorithms for 2D transformations in computer graphics using C programming language. It discusses functions for translation, rotation, and scaling of 2D objects.
The algorithms take input for the type of transformation, parameters for the transformation, and coordinates of the 2D object. Translation simply adds the translation offsets to the x and y coordinates. Rotation rotates the object around the x-axis or y-axis based on the angle input. Scaling multiplies the x and y coordinates by respective scaling factors.
The transformed object is displayed on the graphics screen. Thus basic 2D transformations like translation, rotation, scaling are implemented through these algorithms in C.
Random Number Generation
The document contains solutions to 3 problems involving generating random numbers using different linear congruential methods. The first problem uses the linear congruential method to generate 3 two-digit random integers: 63, 51, and 55. The second problem uses the multiplicative congruential method to generate 4 three-digit random integers: 333, 319, 717, and 831. The third problem uses the mixed congruential method to generate 3 two-digit random numbers: 88, 45, and 44.
IRJET- On Greatest Common Divisor and its Application for a Geometrical S...
This document discusses greatest common divisors (GCD) and their application to geometric structures. It begins by defining GCD and least common multiple (LCM) and proving several properties about them. It then derives formulas for the minimum number of identical rectangular tiles or cuboids needed to construct a square floor or cube, respectively. Specifically, it shows the minimum number of tiles/cuboids is equal to the GCD of certain parameters divided by the area/volume of each tile/cuboid. The document provides examples to illustrate the formulas.
Finite fields
Finite fields are important in cryptography. They involve arithmetic operations on a finite set of elements called a Galois field. Two important examples are GF(p), the integers modulo a prime p, and GF(2n), polynomials modulo an irreducible polynomial of degree n with coefficients in GF(2). Polynomial arithmetic in GF(2n) can be performed using bitwise XOR and shift operations, with modulo reduction using the irreducible polynomial. This allows efficient computation in finite fields important for cryptography.
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
Christian Jett Morales presents on the quadratic function derived from zeros of the function. The presentation includes:
- An introduction to deriving the quadratic function f(x) = ax^2 + bx + c from the zeros r1 and r2 of the function.
- Examples of deriving quadratic functions when given different zeros, such as 3/2 and -1, or 2+√3 and 2-√3.
- An activity for participants to practice deriving quadratic functions.
- A dedication to God and the presenter's teacher for the opportunity to create the presentation.
What's hot (19)
IRJET- On Greatest Common Divisor and its Application for a Geometrical S...
IRJET- On Greatest Common Divisor and its Application for a Geometrical S...
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
Viewers also liked
Calculus II - 25
This document discusses tests for determining if infinite series converge or diverge, including:
1) The Alternating Series Test, which can be used to determine if alternating series (with terms that are alternately positive and negative) converge.
2) Absolute and conditional convergence, where a series is absolutely convergent if the sum of the absolute values converges, and conditionally convergent if the series converges but the sum of absolute values does not.
3) The Ratio and Root Tests, which can be used to determine if series converge absolutely (and thus converge) based on the ratios or roots of successive terms.
Calculus II - 36
The document discusses arc length, curvature, and formulas for calculating them for curves in 2D and 3D. It provides the definitions of arc length, unit tangent vector, and curvature. Examples are given of finding the curvature of various curves, such as circles, helices, and parabolas, at given points using parametrizations of the curves. In general, the curvature of a plane curve f(x) = y is k(x) = f''(x) / [1+(f'(x))^2]^(3/2).
Calculus II - 31
The document discusses 3D coordinate systems and vectors. It defines:
1) The distance formula in 3D space between two points (x1,y1,z1) and (x2,y2,z2) as the square root of (x2-x1)2 + (y2-y1)2 + (z2-z1)2.
2) Vectors as quantities that have both magnitude and direction. It shows examples of vector addition and subtraction using the triangle and parallelogram laws.
3) Key properties of vectors including that vector addition is commutative and distributive over vector addition, scalar multiplication distributes over vector addition, and vectors
Calculus II - 19
There are three types of conic sections: parabolas, ellipses, and hyperbolas. A parabola is the set of points equidistant from a fixed point (focus) and line (directrix). An ellipse is the set of points where the sum of distances from two fixed points (foci) is a constant. A hyperbola is the set of points where the difference of distances from two fixed points (foci) is a constant. Equations of parabolas, ellipses, and hyperbolas can be written and their properties like foci, vertices and asymptotes can be determined from the equations.
Caculus II - 37
The document discusses tangent, normal, and binormal vectors for curves. It defines these vectors for a space curve at a point and explains that they form an orthogonal TNB frame. It then gives examples of finding these vectors and the normal and osculating planes for a helix curve at a specific point.
Calculus II - 20
1) A sequence is an ordered list of numbers written as {an}. A sequence converges to a limit L if the terms an get arbitrarily close to L as n increases.
2) Common properties of convergent sequences include that the sum, difference, and scalar multiple of two convergent sequences is also convergent.
3) Useful theorems include if {an} converges to L and {bn} converges to M, then {an + bn} converges to L + M. If |an| ≤ M for all n and M is a constant, then the sequence converges.
4) Examples of convergent sequences include the sequence of partial sums of a convergent series and the terms of a function evaluated at
Calculus II - 29
The document discusses Taylor series and Maclaurin series. It provides the formulas for determining the coefficients of a Taylor series centered at a point a. It also discusses Taylor's inequality which relates the remainder of a Taylor series to the derivative of the function. Several examples are worked out of finding the Maclaurin series for common functions and proving they represent the functions for all values. Additional power series are derived from fundamental series like e^x, sin(x), and cos(x).
Calculus II - 28
The document discusses Taylor series and how they can be used to represent general functions as power series. It provides the formulas for determining the coefficients in the Taylor series expansion of a function f(x) about a point a. Examples are given of finding the Maclaurin series, which is the Taylor series expansion about x=0, for common functions and determining their radii of convergence. The key steps in proving that the Taylor series equals the original function within the radius of convergence are outlined using Taylor's inequality.
Lesson 7: Vector-valued functions
We define vector-valued functions, whose image is a curve in the plane or space. We also show how to differentiate and integrate them
Calculus II - 30
The document discusses using Taylor series approximations to evaluate expressions. It finds the maximum error of approximating e-0.5 as 1 - 0.5 + 0.5^2, determines the value of x for which this approximation is accurate to within 0.01, and establishes that an 8th degree Taylor polynomial is needed to approximate e^x in [-1,1] with an error less than 0.001.
Calculus II - 26
A power series is a series of the form Σan(x-c)n where an are the coefficients of the series. The sum of the series is a function of x. A power series centered at c or about c is of the form Σan(x-c)n. For a given power series, there are three possibilities for convergence: it converges only at c, converges for all x, or converges when |x-c|<R and diverges when |x-c|>R, where R is the radius of convergence.
Calculus II - 23
The document introduces the integral test, which can be used to determine if an infinite series is convergent or divergent. The integral test states that a series is convergent if the corresponding improper integral converges, and divergent if the improper integral diverges. Several examples demonstrate applying the integral test to various series by setting up and evaluating the corresponding improper integrals. The document also discusses using integrals to estimate the sum of a convergent series.
Calculus II - 21
This document discusses infinite sequences and infinite series. Some key points:
1. It provides examples of determining if an infinite sequence converges or diverges based on the limit of its terms.
2. Properties of operations on convergent sequences are given, such as sums, products, and compositions.
3. Theorems regarding convergence of sequences are stated, such as if the limit of the sequence of partial sums exists.
4. Examples are given of showing monotonicity (increasing or decreasing behavior) of specific sequences.
5. Boundedness of sequences is defined, and the monotonic sequence theorem is stated.
6. Infinite series are introduced as the sum of the terms of a
Calculus II - 18
The document discusses formulas for calculating the area and length of regions defined by polar curves. It gives the formula for finding the area inside a polar curve between rays at specific theta values. Similarly, it provides the formula for calculating the arc length of a polar curve between two theta values. It then works through examples of using these formulas to find the area of a four-leaved rose curve and the area and length of a cardioid curve.
Calculus II - 27
This document discusses power series representations of functions and their radii of convergence. It provides examples of finding the power series for functions like 1/(1+x) and sin(x), as well as the general process for determining the power series representation of a function and the radius of convergence based on the interval of convergence. It also discusses rules for differentiating and integrating term-by-term for power series whose functions have a given radius of convergence.
Calculus II - 24
The Comparison Test allows determining if a series converges or diverges by comparing it to standard p-series or geometric series. If a series is less than or equal to a convergent p-series, or greater than or equal to a divergent p-series, then the series has the same behavior. Examples demonstrate applying the test to various series.
Calculus II - 22
This document discusses convergence of infinite series. It defines partial sums of a series and states that if the sequence of partial sums converges, then the series converges to that limit. If the partial sums do not converge or their limit does not exist, then the series diverges. It provides examples of geometric series, which converge if the common ratio is less than 1 in absolute value, and harmonic series, which diverges. It also discusses using known convergent series to determine if new series formed from their sums or products converge.
1605 power series
This document discusses power series and their intervals and radii of convergence. It begins by introducing power series notation and providing examples. It then defines the interval of convergence as the set of values for which the series converges, and the radius of convergence as half the length of the interval of convergence. The document outlines three possible types of intervals of convergence and provides examples. It concludes by describing the general method to determine a power series' interval of convergence, which involves finding the limit of terms, solving an inequality, and checking endpoint values.
Viewers also liked (18)
More from David Mao
Calculus II - 17
Polar coordinates represent points in a plane using an angle and distance from an origin point, rather than Cartesian x and y distances. In polar coordinates, a point is defined by its radial distance r from the origin and its angle θ measured counterclockwise from the polar axis. Converting between polar and Cartesian coordinates involves simple formulas relating r, θ, x, and y. Polar equations can more simply describe curves like the cardioid than their Cartesian equivalents.
Calculus II - 16
The document discusses calculus concepts related to parametric curves including calculating tangents, areas, arc length, and surfaces of revolution. It provides the formulas for finding the tangent of a parametric curve, area under a parametric curve, arc length of a parametric curve, and surface area obtained by rotating a parametric curve around an axis. Examples are given for each concept using the parametric cycloid curve.
Calculus II - 15
The document discusses parametric equations and parametric curves. Parametric equations define curves using two equations, where x and y are defined in terms of a parameter t. Common examples of parametric curves are given. Parametric equations can be used to define circles by giving the x and y coordinates in terms of the center point and radius. The parameter t can be eliminated to obtain the Cartesian equation of the curve. Calculus can be applied to parametric curves to find tangents, areas, arc length, and surfaces of revolution. Formulas are given for finding the equations of tangent lines to parametric curves at a given value of t.
Calculus II - 14
Separable differential equations can be solved explicitly by factoring the expression for the derivative into a function of y and a function of x. Examples of separable equations provided include dy/dx = x, dy/dx = y + x, and dy/dx = y/x. These types of first order differential equations can be solved by separating the variables and integrating both sides.
Calculus II - 13
This document discusses numerical methods for solving differential equations. It introduces direction fields, which provide a graphical approach to studying solutions, and Euler's method, which provides a numerical approach. Euler's method works by approximating the slope of the tangent line at each step using small step sizes to iteratively calculate successive approximations of the solution.
Calculus II - 12
The document discusses differential equations, which are equations containing an unknown function and one or more of its derivatives. A function is a solution to a differential equation if it satisfies the equation. Common types of differential equations include those modeling population growth over time and the motion of springs. These examples show how differential equations can be used to model real-world phenomena and how initial conditions are used to fully define a unique solution to what is called an initial-value problem.
Calculus II - 11
This document discusses finding the center of mass of objects using calculus. It provides examples of finding the center of mass for discrete objects, continuous objects defined by a function, and regions bounded by curves. The key steps are to calculate the moments about the x- and y-axes and then use these to determine the x- and y-coordinates of the center of mass. Examples include finding the center of a semicircular plate and the centroid of a region bounded by a line and parabola.
Calculus II - 10
The document discusses calculating the area of a surface obtained by rotating a curve around an axis. It provides the general formulas for finding the area which involves integrating the function squared plus the integral of its derivative. Several examples are worked out for rotating curves and arcs around the x-axis and y-axis. The document also gives an example problem of finding the force on a dam due to hydrostatic water pressure.
Calculus II - 9
The document discusses the arc length formula and how to calculate arc length. The arc length formula is the integral of the square root of (dx/dt)^2 + (dy/dt)^2 from t1 to t2, where t1 and t2 are the starting and ending points. Examples are provided on calculating the perimeter of a circle using this formula and approximating the arc length of a hyperbola between two points.
Calculus II - 8
This document discusses different types of improper integrals. It defines Type I improper integrals as integrals where the integrand is defined at every point but the limit at one or both endpoints is infinite. Type II integrals have an integrand that is continuous on some interval excluding one or both endpoints. The document provides examples of evaluating various improper integrals and checking whether they converge or diverge. It also introduces a comparison theorem for improper integrals.
Calculus II - 7
This document discusses numerical integration techniques for approximating integrals of functions that have no closed-form antiderivative. It introduces the midpoint rule, trapezoidal rule, and Simpson's rule for approximating integrals using Riemann sums over subintervals. An example applies Simpson's rule to approximate a definite integral and estimates the error bound.
More from David Mao (11)
Recently uploaded
“Temporal Event Neural Networks: A More Efficient Alternative to the Transfor...
“Temporal Event Neural Networks: A More Efficient Alternative to the Transfor...Edge AI and Vision Alliance
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/temporal-event-neural-networks-a-more-efficient-alternative-to-the-transformer-a-presentation-from-brainchip/
Chris Jones, Director of Product Management at BrainChip , presents the “Temporal Event Neural Networks: A More Efficient Alternative to the Transformer” tutorial at the May 2024 Embedded Vision Summit.
The expansion of AI services necessitates enhanced computational capabilities on edge devices. Temporal Event Neural Networks (TENNs), developed by BrainChip, represent a novel and highly efficient state-space network. TENNs demonstrate exceptional proficiency in handling multi-dimensional streaming data, facilitating advancements in object detection, action recognition, speech enhancement and language model/sequence generation. Through the utilization of polynomial-based continuous convolutions, TENNs streamline models, expedite training processes and significantly diminish memory requirements, achieving notable reductions of up to 50x in parameters and 5,000x in energy consumption compared to prevailing methodologies like transformers.
Integration with BrainChip’s Akida neuromorphic hardware IP further enhances TENNs’ capabilities, enabling the realization of highly capable, portable and passively cooled edge devices. This presentation delves into the technical innovations underlying TENNs, presents real-world benchmarks, and elucidates how this cutting-edge approach is positioned to revolutionize edge AI across diverse applications.[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...
The typical problem in product engineering is not bad strategy, so much as “no strategy”. This leads to confusion, lack of motivation, and incoherent action. The next time you look for a strategy and find an empty space, instead of waiting for it to be filled, I will show you how to fill it in yourself. If you’re wrong, it forces a correction. If you’re right, it helps create focus. I’ll share how I’ve approached this in the past, both what works and lessons for what didn’t work so well.
Harnessing the Power of NLP and Knowledge Graphs for Opioid Research
Gursev Pirge, PhD
Senior Data Scientist - JohnSnowLabs
Energy Efficient Video Encoding for Cloud and Edge Computing Instances
Energy Efficient Video Encoding for Cloud and Edge Computing Instances
The Microsoft 365 Migration Tutorial For Beginner.pptx
This presentation will help you understand the power of Microsoft 365. However, we have mentioned every productivity app included in Office 365. Additionally, we have suggested the migration situation related to Office 365 and how we can help you.
You can also read: https://www.systoolsgroup.com/updates/office-365-tenant-to-tenant-migration-step-by-step-complete-guide/
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor Ivaniuk
At this talk we will discuss DDoS protection tools and best practices, discuss network architectures and what AWS has to offer. Also, we will look into one of the largest DDoS attacks on Ukrainian infrastructure that happened in February 2022. We'll see, what techniques helped to keep the web resources available for Ukrainians and how AWS improved DDoS protection for all customers based on Ukraine experience
Programming Foundation Models with DSPy - Meetup Slides
Prompting language models is hard, while programming language models is easy. In this talk, I will discuss the state-of-the-art framework DSPy for programming foundation models with its powerful optimizers and runtime constraint system.
zkStudyClub - LatticeFold: A Lattice-based Folding Scheme and its Application...
Folding is a recent technique for building efficient recursive SNARKs. Several elegant folding protocols have been proposed, such as Nova, Supernova, Hypernova, Protostar, and others. However, all of them rely on an additively homomorphic commitment scheme based on discrete log, and are therefore not post-quantum secure. In this work we present LatticeFold, the first lattice-based folding protocol based on the Module SIS problem. This folding protocol naturally leads to an efficient recursive lattice-based SNARK and an efficient PCD scheme. LatticeFold supports folding low-degree relations, such as R1CS, as well as high-degree relations, such as CCS. The key challenge is to construct a secure folding protocol that works with the Ajtai commitment scheme. The difficulty, is ensuring that extracted witnesses are low norm through many rounds of folding. We present a novel technique using the sumcheck protocol to ensure that extracted witnesses are always low norm no matter how many rounds of folding are used. Our evaluation of the final proof system suggests that it is as performant as Hypernova, while providing post-quantum security.
Paper Link: https://eprint.iacr.org/2024/257
“How Axelera AI Uses Digital Compute-in-memory to Deliver Fast and Energy-eff...
“How Axelera AI Uses Digital Compute-in-memory to Deliver Fast and Energy-eff...Edge AI and Vision Alliance
For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/how-axelera-ai-uses-digital-compute-in-memory-to-deliver-fast-and-energy-efficient-computer-vision-a-presentation-from-axelera-ai/
Bram Verhoef, Head of Machine Learning at Axelera AI, presents the “How Axelera AI Uses Digital Compute-in-memory to Deliver Fast and Energy-efficient Computer Vision” tutorial at the May 2024 Embedded Vision Summit.
As artificial intelligence inference transitions from cloud environments to edge locations, computer vision applications achieve heightened responsiveness, reliability and privacy. This migration, however, introduces the challenge of operating within the stringent confines of resource constraints typical at the edge, including small form factors, low energy budgets and diminished memory and computational capacities. Axelera AI addresses these challenges through an innovative approach of performing digital computations within memory itself. This technique facilitates the realization of high-performance, energy-efficient and cost-effective computer vision capabilities at the thin and thick edge, extending the frontier of what is achievable with current technologies.
In this presentation, Verhoef unveils his company’s pioneering chip technology and demonstrates its capacity to deliver exceptional frames-per-second performance across a range of standard computer vision networks typical of applications in security, surveillance and the industrial sector. This shows that advanced computer vision can be accessible and efficient, even at the very edge of our technological ecosystem.Apps Break Data
How information systems are built or acquired puts information, which is what they should be about, in a secondary place. Our language adapted accordingly, and we no longer talk about information systems but applications. Applications evolved in a way to break data into diverse fragments, tightly coupled with applications and expensive to integrate. The result is technical debt, which is re-paid by taking even bigger "loans", resulting in an ever-increasing technical debt. Software engineering and procurement practices work in sync with market forces to maintain this trend. This talk demonstrates how natural this situation is. The question is: can something be done to reverse the trend?
Digital Banking in the Cloud: How Citizens Bank Unlocked Their Mainframe
Inconsistent user experience and siloed data, high costs, and changing customer expectations – Citizens Bank was experiencing these challenges while it was attempting to deliver a superior digital banking experience for its clients. Its core banking applications run on the mainframe and Citizens was using legacy utilities to get the critical mainframe data to feed customer-facing channels, like call centers, web, and mobile. Ultimately, this led to higher operating costs (MIPS), delayed response times, and longer time to market.
Ever-changing customer expectations demand more modern digital experiences, and the bank needed to find a solution that could provide real-time data to its customer channels with low latency and operating costs. Join this session to learn how Citizens is leveraging Precisely to replicate mainframe data to its customer channels and deliver on their “modern digital bank” experiences.
June Patch Tuesday
Ivanti’s Patch Tuesday breakdown goes beyond patching your applications and brings you the intelligence and guidance needed to prioritize where to focus your attention first. Catch early analysis on our Ivanti blog, then join industry expert Chris Goettl for the Patch Tuesday Webinar Event. There we’ll do a deep dive into each of the bulletins and give guidance on the risks associated with the newly-identified vulnerabilities.
Your One-Stop Shop for Python Success: Top 10 US Python Development Providers
Simplify your search for a reliable Python development partner! This list presents the top 10 trusted US providers offering comprehensive Python development services, ensuring your project's success from conception to completion.
5th LF Energy Power Grid Model Meet-up Slides
5th Power Grid Model Meet-up
It is with great pleasure that we extend to you an invitation to the 5th Power Grid Model Meet-up, scheduled for 6th June 2024. This event will adopt a hybrid format, allowing participants to join us either through an online Mircosoft Teams session or in person at TU/e located at Den Dolech 2, Eindhoven, Netherlands. The meet-up will be hosted by Eindhoven University of Technology (TU/e), a research university specializing in engineering science & technology.
Power Grid Model
The global energy transition is placing new and unprecedented demands on Distribution System Operators (DSOs). Alongside upgrades to grid capacity, processes such as digitization, capacity optimization, and congestion management are becoming vital for delivering reliable services.
Power Grid Model is an open source project from Linux Foundation Energy and provides a calculation engine that is increasingly essential for DSOs. It offers a standards-based foundation enabling real-time power systems analysis, simulations of electrical power grids, and sophisticated what-if analysis. In addition, it enables in-depth studies and analysis of the electrical power grid’s behavior and performance. This comprehensive model incorporates essential factors such as power generation capacity, electrical losses, voltage levels, power flows, and system stability.
Power Grid Model is currently being applied in a wide variety of use cases, including grid planning, expansion, reliability, and congestion studies. It can also help in analyzing the impact of renewable energy integration, assessing the effects of disturbances or faults, and developing strategies for grid control and optimization.
What to expect
For the upcoming meetup we are organizing, we have an exciting lineup of activities planned:
-Insightful presentations covering two practical applications of the Power Grid Model.
-An update on the latest advancements in Power Grid -Model technology during the first and second quarters of 2024.
-An interactive brainstorming session to discuss and propose new feature requests.
-An opportunity to connect with fellow Power Grid Model enthusiasts and users.
Main news related to the CCS TSI 2023 (2023/1695)
An English 🇬🇧 translation of a presentation to the speech I gave about the main changes brought by CCS TSI 2023 at the biggest Czech conference on Communications and signalling systems on Railways, which was held in Clarion Hotel Olomouc from 7th to 9th November 2023 (konferenceszt.cz). Attended by around 500 participants and 200 on-line followers.
The original Czech 🇨🇿 version of the presentation can be found here: https://www.slideshare.net/slideshow/hlavni-novinky-souvisejici-s-ccs-tsi-2023-2023-1695/269688092 .
The videorecording (in Czech) from the presentation is available here: https://youtu.be/WzjJWm4IyPk?si=SImb06tuXGb30BEH .
Mutation Testing for Task-Oriented Chatbots
Conversational agents, or chatbots, are increasingly used to access all sorts of services using natural language. While open-domain chatbots - like ChatGPT - can converse on any topic, task-oriented chatbots - the focus of this paper - are designed for specific tasks, like booking a flight, obtaining customer support, or setting an appointment. Like any other software, task-oriented chatbots need to be properly tested, usually by defining and executing test scenarios (i.e., sequences of user-chatbot interactions). However, there is currently a lack of methods to quantify the completeness and strength of such test scenarios, which can lead to low-quality tests, and hence to buggy chatbots.
To fill this gap, we propose adapting mutation testing (MuT) for task-oriented chatbots. To this end, we introduce a set of mutation operators that emulate faults in chatbot designs, an architecture that enables MuT on chatbots built using heterogeneous technologies, and a practical realisation as an Eclipse plugin. Moreover, we evaluate the applicability, effectiveness and efficiency of our approach on open-source chatbots, with promising results.
How to Interpret Trends in the Kalyan Rajdhani Mix Chart.pdf
A Mix Chart displays historical data of numbers in a graphical or tabular form. The Kalyan Rajdhani Mix Chart specifically shows the results of a sequence of numbers over different periods.
Recently uploaded (20)
“Temporal Event Neural Networks: A More Efficient Alternative to the Transfor...
“Temporal Event Neural Networks: A More Efficient Alternative to the Transfor...
[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...
[OReilly Superstream] Occupy the Space: A grassroots guide to engineering (an...
Harnessing the Power of NLP and Knowledge Graphs for Opioid Research
Harnessing the Power of NLP and Knowledge Graphs for Opioid Research
Energy Efficient Video Encoding for Cloud and Edge Computing Instances
Energy Efficient Video Encoding for Cloud and Edge Computing Instances
The Microsoft 365 Migration Tutorial For Beginner.pptx
The Microsoft 365 Migration Tutorial For Beginner.pptx
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor Ivaniuk
"Frontline Battles with DDoS: Best practices and Lessons Learned", Igor Ivaniuk
Programming Foundation Models with DSPy - Meetup Slides
Programming Foundation Models with DSPy - Meetup Slides
zkStudyClub - LatticeFold: A Lattice-based Folding Scheme and its Application...
zkStudyClub - LatticeFold: A Lattice-based Folding Scheme and its Application...
“How Axelera AI Uses Digital Compute-in-memory to Deliver Fast and Energy-eff...
“How Axelera AI Uses Digital Compute-in-memory to Deliver Fast and Energy-eff...
Nordic Marketo Engage User Group_June 13_ 2024.pptx
Nordic Marketo Engage User Group_June 13_ 2024.pptx
Digital Banking in the Cloud: How Citizens Bank Unlocked Their Mainframe
Digital Banking in the Cloud: How Citizens Bank Unlocked Their Mainframe
Deep Dive: AI-Powered Marketing to Get More Leads and Customers with HyperGro...
Deep Dive: AI-Powered Marketing to Get More Leads and Customers with HyperGro...
Your One-Stop Shop for Python Success: Top 10 US Python Development Providers
Your One-Stop Shop for Python Success: Top 10 US Python Development Providers
How to Interpret Trends in the Kalyan Rajdhani Mix Chart.pdf
How to Interpret Trends in the Kalyan Rajdhani Mix Chart.pdf
Overcoming the PLG Trap: Lessons from Canva's Head of Sales & Head of EMEA Da...
Overcoming the PLG Trap: Lessons from Canva's Head of Sales & Head of EMEA Da...
Calculus II - 33
- 1. 12.3 The Dot Product The dot product (or inner product) of = , , and = , , is given by · = + + It can also be defined for 2D vectors. Ex: , , · , , = · + ( )+ · = + · = · + · +( )( )=
- 2. Properties (I): · = · ·( + )= · + · ( )· = ( · )= ·( ) · =
- 3. Properties (II): · =| | · = | || | · =
- 4. Properties (II): · =| | · = | || | · = · > · = · <
- 5. Direction Angles and Direction Cosines The direction angles of a nonzero vector are the angles that it makes with the positive x-, y-, and z-axes. · · = = = = | || | | | | || | | | · = = | || | | | + + = =| | , ,
- 6. Projections: The scalar projection of onto (also called the component of onto ) is defined to be · =| | = calar! | | sig ne d s The projection of onto is defined to be · · = = ve ctor! | | | | | |
- 7. 12.4 The Cross Product The cross product of = , , and = , , is given by = , , = = + It can only be defined for 3D vectors.
- 8. Properties (I): = ( + )= + ( ) = ( )= ( ) = · = · · )= · + ·( + ·( ) = ( · )= ( )· · =
- 9. Properties (II): = | | = | || | = · =| | · = | || | · =
- 10. Properties (III): ( ) ,( ) | | = | || | equals to the area of the parallelogram determined by and .
- 11. Properties (III): ( ) ,( ) | | = | || | equals to the area of the parallelogram determined by and . The Right Hand Rule: If the fingers of your right hand curl in the direction of a rotation from to , then your thumb points in the direction of .
Editor's Notes
- \n
- \n
- \n
- \n
- \n
- \n
- \n
- \n
- \n
- \n