28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z


 Marian Jenkins
 5 years ago
 Views:
Transcription
1 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition of this new operation is that we wanted its result to be a vector, perpendicular to the two vectors we are taking the cross product of. Unlike the dot product, the cross product is only defined for 3D vectors. In this section, when we use the word vector, we will mean 3D vector. Definition 45 (cross product) The cross product also called vector product of two vectors u = u x, u y, u z and v = v x, v y, v z, denoted u v, is defined to be u x v x u y v z u z v y u y u z v y v z = u z v x u x v z u x v y u y v x Thus, the cross product of two 3D vectors is also a 3D vector. This formula is not easy to remember. However, if you know about matrices and the determinant of a matrix, the cross product can be expressed in term of them. Let us first quickly review what they are. Definition 46 We only give the definition of the determinant of a 2 2 and a 3 3 matrix. [ ] a b 1. The determinant of a 2 2 matrix, denoted by c d a b c d is defined to be a b c d = ad bc a 1 a 2 a 3 a 1 a 2 a 3 2. The determinant of a 3 3 matrix b 1 b 2 b 3 denoted by b 1 b 2 b 3 c 1 c 2 c 3 c 1 c 2 c 3 is defined to be a 1 a 2 a 3 b 1 b 2 b 3 b = a 2 b 3 1 c 1 c 2 c 3 c 2 c 3 a 2 b 1 b 3 c 1 c 3 + a 3 b 1 b 2 c 1 c 2 = a 1 (b 2 c 3 c 2 b 3 ) a 2 (b 1 c 3 c 1 b 3 ) + a 3 (b 1 c 2 c 1 b 2 ) Example 47 Find = (1) (3) (7) (2) = 3 14 = 11
2 1.4. CROSS PRODUCT 29 Example 48 Find = = (1) (2 7) 2 (6 4) + 3 (21 4) = = 42 Proposition 49 If u = u x, u y, u z and v = v x, v y, v z then i j k u v = u x u y u z v x v y v z Which makes it much easier to remember. Proof. i j k u x u y u z v x v y v z = (u y v z u z v y ) i (u x v z u z v x ) j + (u x v y u y v x ) k = (u y v z u z v y, u z v x u x v z, u x v y u y v x ) = u v Example 50 For u = 3, 1, 1 and v = 4, 7, 2, compute u v. i j k u v = = (2 7) i (6 4) j + (21 4) k = 5, 2, 17 The above tells us how to compute the cross product. However, it does not tell us what the cross product represents. There is a very nice geometric interpretation of the interpretation of the cross product Properties Theorem 51 Let u and v denote two nonzero vectors. Then, the following is true: 1. u u = 0
3 30 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 2. u v is perpendicular to both u and v. 3. u v = u v sin α where α is the smallest angle between u and v (0 α π). 4. u v = u v sin α n where n is the unit vector perpendicular to both u and v whose direction is determined by the righthand rule. Remark 52 The above properties tell us that u v is the vector perpendicular to both u and v which direction is given by the righthand rule and whose magnitude is u v sin α. This is very important. There are many situations in which one needs to find a vector perpendicular to two known vectors. We illustrate it with examples. Example 53 Find a unit vector perpendicular to both u = 1, 1, 1 and v = 2, 1, 0. First, we find a vector perpendicular to these two vectors, then we make it into a unit vector. Such a vector is A unit vector in the same direction is u v = 1, 1, 1 2, 1, 0 i j k = = 2 j k i = 1, 2, 1 1, 2, 1 1, 2, 1 = \ = 1, 1, , 2, Example 54 Find a vector perpendicular to the plane containing the three points P : (1, 1, 2), Q : (2, 1, 1) and R : (2, 1, 0). As long as the three points are not collinear, we can make two not parallel vectors from them, for example P Q and QR. A vector perpendicular to the plane will be perpendicular to both vectors. Such a vector is P Q QR. P Q = 1, 0, 1 and Therefore, P Q QR = QR = 0, 0, 1 = j i j k = 0, 1, 0
4 1.4. CROSS PRODUCT 31 Remark 55 Using the definition, it is easy to verify that and i j = k j k = i k i = j j i = k k j = i i k = j Remark 56 From property 3 of theorem 51, it follows that two nonzero vectors are parallel if and only if their cross product is 0. The cross product satisfies more properties which we will not prove because they are very tedious. Theorem 57 Let u, v, and w be three vectors and a be a scalar. The following is true: 1. u v = v u (this tells us that the cross product is not commutative. 2. (a u) v = a ( u v) = u (a v) 3. u ( v + w) = u v + u w 4. ( u + v) w = u w + v w Area of a Parallelogram Consider a parallelogram whose sides are given by the vectors u and v as shown in figure Remembering that the area of a parallelogram is the length of its base times its height, we see that the area A of this parallelogram is A = u v sin θ = u v Example 58 Find the area of the parallelogram shown in figure If we let A = (0, 0, 0), B = (0, 4, 0), C = (0, 1, 3) and D = (0, 5, 3) then Area = AB AC First, we compute the cross product. AB AC = 0, 4, 0 0, 1, 3 = 12, 0, 0
5 32 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.13: Area of the paralelogram is u v = u v sin θ Therefore Area = 12, 0, 0 = Triple Products Definition 59 Given three nonzero vectors u, v, and w, the product u ( v w) is called the scalar triple product of the vectors u, v, and w. Proposition 60 The volume of the parallelepiped determined by the vectors u, v, and w as shown in figure 1.15 is the magnitude of their scalar triple product u ( v w). Proof. The volume V of a parallelepiped is given by V = area of the base times height Suppose the base of the parallelepiped is determined by v and w. Let θ be the angle u makes with the direction perpendicular to the base. Then the height of the parallelepiped is u cos θ. The area of the base is v w. Therefore, V = v w u cos θ = u ( v w)
6 1.4. CROSS PRODUCT 33 Figure 1.14: Find the area Figure 1.15: Parallelepiped determined by u, v and w
7 34 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Corollary 61 Three nonzero vectors u, v, and w are coplanar (on the same plane) if u ( v w) = 0. Remark 62 If instead of thinking of the parallelepiped as having its base determined by v and w, we had thought of it as having its base determined by u and v, then we would have found that its volume was w ( u v). But since we are talking about the same parallelepiped, the two formulas for the volume must be the same, so we have: u ( v w) = w ( u v) (1.7) Remark 63 The scalar triple product of three nonzero vectors u, v, and w can be computed by calculating the determinant u x u y u z u ( v w) = v x v y v z (1.8) w x w y w z Summary The cross product is a very important quantity in mathematics. It can be used for: 1. Find a vector perpendicular to two nonzero vectors (often used in computer graphics). 2. Find the area of a parallelogram. 3. Find the volume of a parallelepiped. 4. Determine if two nonzero vectors are parallel. 5. Determine if three nonzero vectors are coplanar. 6. Many applications in physics which we will not discuss here Vectors and Maple To handle vectors using Maple 9.5, one must first load the LinearAlgebra package with the command with(linearalgebra); Once this package is loaded, the following operations can be performed: Defining a vector: This is done using the construct,,. For example, 1 to define the vector A to be 3 4, use A := 1, 3, 4 ;
8 1.4. CROSS PRODUCT 35 Adding two vectors: Use the usual addition symbol as in A + B Scalar Multiplication: Use the usual multiplication symbol as in 2 A Subtracting two vectors: Use the usual subtraction symbol as in A B Finding the norm of a vector: The norm we defined in this class is called the 2norm in more advanced mathematics classes because we take the square root of the sum of the squares of the coordinates. To do this with Maple, use Norm(A, 2); where A is a vector. Dot product: Given two vectors A and B, their dot product can be found using DotProduct(A,B); or the shortcut A.B; Cross product: Given two vectors A and B, their cross product can be found using CrossProduct(A,B); or the shortcut A &x B; There must be spaces between A and & as well as between x and B. Plotting vectors: To plot vectors, one must first load the plots package with the command with(plots); To plot the vector A, one would then use arrow(a, shape=arrow); The shape parameter is optional. To plot two or more vectors, one must list the vectors inside square brackets. The command is: arrow([a,b],shape=arrow); To find all the parameters of the arrow command, use the help facility of Maple.
9 36 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Problems 1. Find the length when defined of u v and v u for the given u and v below. (a) u = 2 i 2 j k and v = i k. (b) u = 2 i 2 j + 4 k and v = i + j 2 k. (c) u = 2 i and v = 3 j. (d) u = 8 i 2 j 4 k and v = 2 i + 2 j + k. 2. Sketch u, v and u u for u = i and v = j. 3. Find the area of the triangle determined by P = (1, 1, 2), Q = (2, 0, 1) and R = (0, 2, 1). Then, find a unit vector perpendicular to the plane P QR. 4. Find the area of the triangle determined by P = (2, 2, 1), Q = (3, 1, 2) and R = (3, 1, 1). Then, find a unit vector perpendicular to the plane P QR. 5. Verify that ( u v ) w = ( v w ) u = ( w u ) v and find the volume of the parallelepiped determined by these three vectors for u = (2, 0, 0), v = (0, 2, 0) and w = (0, 0, 2). 6. Verify that ( u v ) w = ( v w ) u = ( w u ) v and find the volume of the parallelepiped determined by these three vectors for u = (2, 1, 0), v = (2, 1, 1) and w = (1, 0, 2). 7. Let u = 5 i 1 j + k, v = j 5 k, and w = 15 i + 3 j 3 k. Which vectors if any are parallel, perpendicular? 8. Which of the following are always true and which of the following are not always true? Give reasons. (a) u = u u. (b) u u = u. (c) u 0 = 0 u = 0. (d) u u = 0. (e) u v = v u. (f) u ( v + w ) = u v + u w. (g) ( u v ) v = 0. (h) ( u v ) w = u ( v w ). 9. Given nonzero vectors u, v and w, use dot product and cross product notation to describe the following:
10 1.4. CROSS PRODUCT 37 (a) The vector projection of u onto v. (b) A vector orthogonal to u and v. (c) A vector orthogonal to u v and w. (d) The volume of the parallelepiped determined by u, v and w. (e) A vector orthogonal to u v and u w. (f) A vector of length u in the direction of v. 10. Let u, v and w be vectors. Decide which expressions below make sense and which do not. Give reasons. (a) ( u v ) w. (b) u ( v w ). (c) u ( v w ). (d) u ( v w ). 11. Cancellation law. If u v = u w and u 0, does it follow that v = w? Justify your answer. 12. Find the area of the parallelogram whose vertices are A = (1, 0), B = (0, 1), C = ( 1, 0) and D = (0, 1). 13. Find the area of the parallelogram whose vertices are A = ( 1, 2), B = (2, 0), C = (7, 1) and D = (4, 3). 14. Find the area of the triangle whose vertices are A = (0, 0), B = ( 2, 3) and C = (3, 1) 15. Find the area of the triangle whose vertices are A = ( 5, 3), B = (1, 2) and C = (6, 2)
MAT 1341: REVIEW II SANGHOON BAEK
MAT 1341: REVIEW II SANGHOON BAEK 1. Projections and Cross Product 1.1. Projections. Definition 1.1. Given a vector u, the rectangular (or perpendicular or orthogonal) components are two vectors u 1 and
More informationCross product and determinants (Sect. 12.4) Two main ways to introduce the cross product
Cross product and determinants (Sect. 12.4) Two main ways to introduce the cross product Geometrical definition Properties Expression in components. Definition in components Properties Geometrical expression.
More information5.3 The Cross Product in R 3
53 The Cross Product in R 3 Definition 531 Let u = [u 1, u 2, u 3 ] and v = [v 1, v 2, v 3 ] Then the vector given by [u 2 v 3 u 3 v 2, u 3 v 1 u 1 v 3, u 1 v 2 u 2 v 1 ] is called the cross product (or
More informationv 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)
0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3space. This time the outcome will be a vector in 3space. Definition
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationProblem set on Cross Product
1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j  3 k ) 2 Calculate the vector product of i  j and i + j (Ans ) 3 Find the unit vectors that are perpendicular
More informationGeometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi
Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in threedimensional space, we also examine the
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a righthanded set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationMath 241, Exam 1 Information.
Math 241, Exam 1 Information. 9/24/12, LC 310, 11:1512:05. Exam 1 will be based on: Sections 12.112.5, 14.114.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More information6. Vectors. 1 20092016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationThe Dot and Cross Products
The Dot and Cross Products Two common operations involving vectors are the dot product and the cross product. Let two vectors =,, and =,, be given. The Dot Product The dot product of and is written and
More informationMathematics Notes for Class 12 chapter 10. Vector Algebra
1 P a g e Mathematics Notes for Class 12 chapter 10. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. Magnitude of a vector a is denoted by a or a. It is nonnegative
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationOne advantage of this algebraic approach is that we can write down
. Vectors and the dot product A vector v in R 3 is an arrow. It has a direction and a length (aka the magnitude), but the position is not important. Given a coordinate axis, where the xaxis points out
More informationThe Vector or Cross Product
The Vector or ross Product 1 ppendix The Vector or ross Product We saw in ppendix that the dot product of two vectors is a scalar quantity that is a maximum when the two vectors are parallel and is zero
More informationVectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.
Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type
More information9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes
The Scalar Product 9.4 Introduction There are two kinds of multiplication involving vectors. The first is known as the scalar product or dot product. This is socalled because when the scalar product of
More informationVECTOR ALGEBRA. 10.1.1 A quantity that has magnitude as well as direction is called a vector. is given by a and is represented by a.
VECTOR ALGEBRA Chapter 10 101 Overview 1011 A quantity that has magnitude as well as direction is called a vector 101 The unit vector in the direction of a a is given y a and is represented y a 101 Position
More information13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.
3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in threespace, we write a vector in terms
More information9 Multiplication of Vectors: The Scalar or Dot Product
Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation
More informationGeometric description of the cross product of the vectors u and v. The cross product of two vectors is a vector! u x v is perpendicular to u and v
12.4 Cross Product Geometric description of the cross product of the vectors u and v The cross product of two vectors is a vector! u x v is perpendicular to u and v The length of u x v is uv u v sin The
More informationVector Math Computer Graphics Scott D. Anderson
Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationVector Algebra CHAPTER 13. Ü13.1. Basic Concepts
CHAPTER 13 ector Algebra Ü13.1. Basic Concepts A vector in the plane or in space is an arrow: it is determined by its length, denoted and its direction. Two arrows represent the same vector if they have
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationMechanics 1: Vectors
Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized
More informationModule 8 Lesson 4: Applications of Vectors
Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems
More informationLinear Algebra Notes for Marsden and Tromba Vector Calculus
Linear Algebra Notes for Marsden and Tromba Vector Calculus ndimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationChapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
More informationLinear Algebra: Vectors
A Linear Algebra: Vectors A Appendix A: LINEAR ALGEBRA: VECTORS TABLE OF CONTENTS Page A Motivation A 3 A2 Vectors A 3 A2 Notational Conventions A 4 A22 Visualization A 5 A23 Special Vectors A 5 A3 Vector
More informationReview A: Vector Analysis
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A0 A.1 Vectors A2 A.1.1 Introduction A2 A.1.2 Properties of a Vector A2 A.1.3 Application of Vectors
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals,
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More informationα = u v. In other words, Orthogonal Projection
Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v
More informationAP Physics  Vector Algrebra Tutorial
AP Physics  Vector Algrebra Tutorial Thomas Jefferson High School for Science and Technology AP Physics Team Summer 2013 1 CONTENTS CONTENTS Contents 1 Scalars and Vectors 3 2 Rectangular and Polar Form
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationVectors Math 122 Calculus III D Joyce, Fall 2012
Vectors Math 122 Calculus III D Joyce, Fall 2012 Vectors in the plane R 2. A vector v can be interpreted as an arro in the plane R 2 ith a certain length and a certain direction. The same vector can be
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationex) What is the component form of the vector shown in the picture above?
Vectors A ector is a directed line segment, which has both a magnitude (length) and direction. A ector can be created using any two points in the plane, the direction of the ector is usually denoted by
More informationMATH 275: Calculus III. Lecture Notes by Angel V. Kumchev
MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. ThreeDimensional Coordinate Systems..................... 1 Lecture
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More informationGCE Mathematics (6360) Further Pure unit 4 (MFP4) Textbook
Version 36 klm GCE Mathematics (636) Further Pure unit 4 (MFP4) Textbook The Assessment and Qualifications Alliance (AQA) is a company limited by guarantee registered in England and Wales 364473 and a
More informationThe Geometry of the Dot and Cross Products
Journal of Online Mathematics and Its Applications Volume 6. June 2006. Article ID 1156 The Geometry of the Dot and Cross Products Tevian Dray Corinne A. Manogue 1 Introduction Most students first learn
More information2 Session Two  Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision  Session 2: Complex Numbers and Vectors 1 2 Session Two  Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationThe Geometry of the Dot and Cross Products
The Geometry of the Dot and Cross Products Tevian Dray Department of Mathematics Oregon State University Corvallis, OR 97331 tevian@math.oregonstate.edu Corinne A. Manogue Department of Physics Oregon
More informationarxiv:1404.6042v1 [math.dg] 24 Apr 2014
Angle Bisectors of a Triangle in Lorentzian Plane arxiv:1404.604v1 [math.dg] 4 Apr 014 Joseph Cho August 5, 013 Abstract In Lorentzian geometry, limited definition of angles restricts the use of angle
More informationSection 9.1 Vectors in Two Dimensions
Section 9.1 Vectors in Two Dimensions Geometric Description of Vectors A vector in the plane is a line segment with an assigned direction. We sketch a vector as shown in the first Figure below with an
More informationVectors VECTOR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the vector product of two vectors. Table of contents Begin Tutorial
Vectors VECTOR PRODUCT Graham S McDonald A Tutorial Module for learning about the vector product of two vectors Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk 1. Theory 2. Exercises
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationSolutions to Practice Problems
Higher Geometry Final Exam Tues Dec 11, 57:30 pm Practice Problems (1) Know the following definitions, statements of theorems, properties from the notes: congruent, triangle, quadrilateral, isosceles
More informationUseful Mathematical Symbols
32 Useful Mathematical Symbols Symbol What it is How it is read How it is used Sample expression + * ddition sign OR Multiplication sign ND plus or times and x Multiplication sign times Sum of a few disjunction
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationSolving Simultaneous Equations and Matrices
Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering
More information5 VECTOR GEOMETRY. 5.0 Introduction. Objectives. Activity 1
5 VECTOR GEOMETRY Chapter 5 Vector Geometry Objectives After studying this chapter you should be able to find and use the vector equation of a straight line; be able to find the equation of a plane in
More informationOrthogonal Projections and Orthonormal Bases
CS 3, HANDOUT A, 3 November 04 (adjusted on 7 November 04) Orthogonal Projections and Orthonormal Bases (continuation of Handout 07 of 6 September 04) Definition (Orthogonality, length, unit vectors).
More informationOrthogonal Projections
Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationDefinition: A vector is a directed line segment that has and. Each vector has an initial point and a terminal point.
6.1 Vectors in the Plane PreCalculus 6.1 VECTORS IN THE PLANE Learning Targets: 1. Find the component form and the magnitude of a vector.. Perform addition and scalar multiplication of two vectors. 3.
More informationDr. Fritz Wilhelm, DVC,8/30/2004;4:25 PM E:\Excel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM
E:\Ecel files\ch 03 Vector calculations.doc Last printed 8/30/2004 4:25:00 PM Vector calculations 1 of 6 Vectors are ordered sequences of numbers. In three dimensions we write vectors in an of the following
More informationGRADES 7, 8, AND 9 BIG IDEAS
Table 1: Strand A: BIG IDEAS: MATH: NUMBER Introduce perfect squares, square roots, and all applications Introduce rational numbers (positive and negative) Introduce the meaning of negative exponents for
More informationPhysics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus
Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationx1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.
Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationProblem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.
Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve
More informationVector Algebra II: Scalar and Vector Products
Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define
More informationSection 4.4 Inner Product Spaces
Section 4.4 Inner Product Spaces In our discussion of vector spaces the specific nature of F as a field, other than the fact that it is a field, has played virtually no role. In this section we no longer
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationBALTIC OLYMPIAD IN INFORMATICS Stockholm, April 1822, 2009 Page 1 of?? ENG rectangle. Rectangle
Page 1 of?? ENG rectangle Rectangle Spoiler Solution of SQUARE For start, let s solve a similar looking easier task: find the area of the largest square. All we have to do is pick two points A and B and
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More information83 Dot Products and Vector Projections
83 Dot Products and Vector Projections Find the dot product of u and v Then determine if u and v are orthogonal 1u =, u and v are not orthogonal 2u = 3u =, u and v are not orthogonal 6u = 11i + 7j; v
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More information4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
More informationMTH4103: Geometry I. Dr John N. Bray, Queen Mary, University of London
MTH4103: Geometry I Dr John N Bray, Queen Mary, University of London January March 2014 Contents Preface iv 1 Vectors 1 11 Introduction 1 12 Vectors 2 13 The zero vector 3 14 Vector negation 3 15 Parallelograms
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationElementary Linear Algebra
Elementary Linear Algebra Kuttler January, Saylor URL: http://wwwsaylororg/courses/ma/ Saylor URL: http://wwwsaylororg/courses/ma/ Contents Some Prerequisite Topics Sets And Set Notation Functions Graphs
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More informationGeometry: Unit 1 Vocabulary TERM DEFINITION GEOMETRIC FIGURE. Cannot be defined by using other figures.
Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More information... ... . (2,4,5).. ...
12 Three Dimensions ½¾º½ Ì ÓÓÖ Ò Ø ËÝ Ø Ñ So far wehave been investigatingfunctions ofthe form y = f(x), withone independent and one dependent variable Such functions can be represented in two dimensions,
More information