# 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

Save this PDF as:

Size: px
Start display at page:

Download "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"

## 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 3-D vectors. In this section, when we use the word vector, we will mean 3-D 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 3-D vectors is also a 3-D 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 non-zero 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 right-hand 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 right-hand 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 non-zero 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 non-zero 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 non-zero 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 non-zero 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 non-zero vectors (often used in computer graphics). 2. Find the area of a parallelogram. 3. Find the volume of a parallelepiped. 4. Determine if two non-zero vectors are parallel. 5. Determine if three non-zero 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 2-norm 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)

### Cross 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.

### 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

### 5.3 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

### 1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two non-zero 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

### Dot 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

### v 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 3-space. This time the outcome will be a vector in 3-space. Definition

### Geometry 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 three-dimensional space, we also examine the

### Notice that v v w (4)( 15) ( 3)( 20) (0)(2) ( 2)( 15) (2)( 20) (5)(2)

The Cross Product When discussing the dot product, we showed how two vectors can be combined to get a number. Here, we shall see another way of combining vectors, this time resulting in a vector. This

### Problem 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

### v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed 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

### Solution: 2. Sketch the graph of 2 given the vectors and shown below.

7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit

### Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis

Chapter 5 Polar Coordinates; Vectors 5.1 Polar coordinates 1. Pole and polar axis 2. Polar coordinates A point P in a polar coordinate system is represented by an ordered pair of numbers (r, θ). If r >

### Math 241, Exam 1 Information.

Math 241, Exam 1 Information. 9/24/12, LC 310, 11:15-12:05. Exam 1 will be based on: Sections 12.1-12.5, 14.1-14.3. The corresponding assigned homework problems (see http://www.math.sc.edu/ boylan/sccourses/241fa12/241.html)

### 6. Vectors. 1 2009-2016 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,

### 13.4 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

### a a. θ = cos 1 a b ) b For non-zero vectors a and b, then the component of b along a is given as comp

Textbook Assignment 4 Your Name: LAST NAME, FIRST NAME (YOUR STUDENT ID: XXXX) Your Instructors Name: Prof. FIRST NAME LAST NAME YOUR SECTION: MATH 0300 XX Due Date: NAME OF DAY, MONTH DAY, YEAR. SECTION

### One 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 x-axis points out

### Section 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

### Mathematics 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 non-negative

### Lecture 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

### Unified 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,

### Vectors, Gradient, Divergence and Curl.

Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use

### Vectors 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

### The 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

### Figure 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

### VECTOR 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

### Section 1.2. Angles and the Dot Product. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.2 Angles and the Dot Product Suppose x = (x 1, x 2 ) and y = (y 1, y 2 ) are two vectors in R 2, neither of which is the zero vector 0. Let α and

### 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

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

### A 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

### 9 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

### Adding 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

### MATH 304 Linear Algebra Lecture 24: Scalar product.

MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent

### Equations 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

### ELEMENTS OF VECTOR ALGEBRA

ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions

### The 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

### Determinants, Areas and Volumes

Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a two-dimensional object such as a region of the plane and the volume of a three-dimensional object such as a solid

### Vector 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

### Practice Problems for Midterm 1

Practice Problems for Midterm 1 Here are some problems for you to try. A few I made up, others I found from a variety of sources, including Stewart s Multivariable Calculus book. (1) A boy throws a football

### THREE 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,

### Vector 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

### 9.4. 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 so-called because when the scalar product of

### 13 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 three-space, we write a vector in terms

### Vectors. Philippe B. Laval. Spring 2012 KSU. Philippe B. Laval (KSU) Vectors Spring /

Vectors Philippe B Laval KSU Spring 2012 Philippe B Laval (KSU) Vectors Spring 2012 1 / 18 Introduction - Definition Many quantities we use in the sciences such as mass, volume, distance, can be expressed

### Mechanics 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

### Module 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

### Lectures 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 real-valued matrix A is said to be an orthogonal

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### MA261-A Calculus III 2006 Fall Homework 2 Solutions Due 9/13/2006 8:00AM

MA6-A Calculus III 006 Fall Homework Solutions Due 9/3/006 8:00AM 93 #6 Find a b, where a hs; s; 3si and b ht; t; 5ti a b (s) (t) + (s) ( 93 #8 Find a b, where a 4j 3k and b i + 4j + 6k t) + (3s) (5t)

### Vectors-Algebra and Geometry

Chapter Two Vectors-Algebra and Geometry 21 Vectors A directed line segment in space is a line segment together with a direction Thus the directed line segment from the point P to the point Q is different

### α = 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

### Chapter 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

### Linear 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

### FURTHER 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: -9-7 Mathematics

### Vectors What are Vectors? which measures how far the vector reaches in each direction, i.e. (x, y, z).

1 1. What are Vectors? A vector is a directed line segment. A vector can be described in two ways: Component form Magnitude and Direction which measures how far the vector reaches in each direction, i.e.

### LINEAR 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,

### We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus 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

### The Dot Product. If v = a 1 i + b 1 j and w = a 2 i + b 2 j are vectors then their dot product is given by: v w = a 1 a 2 + b 1 b 2

The Dot Product In this section, e ill no concentrate on the vector operation called the dot product. The dot product of to vectors ill produce a scalar instead of a vector as in the other operations that

### Vector algebra Christian Miller CS Fall 2011

Vector algebra Christian Miller CS 354 - Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

### Review A: Vector Analysis

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A-0 A.1 Vectors A-2 A.1.1 Introduction A-2 A.1.2 Properties of a Vector A-2 A.1.3 Application of Vectors

### M243. Fall Homework 2. Solutions.

M43. Fall 011. Homework. s. H.1 Given a cube ABCDA 1 B 1 C 1 D 1, with sides AA 1, BB 1, CC 1 and DD 1 being parallel (can think of them as vertical ). (i) Find the angle between diagonal AC 1 of a cube

### Lecture 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

### Linear Algebra Test 2 Review by JC McNamara

Linear Algebra Test 2 Review by JC McNamara 2.3 Properties of determinants: det(a T ) = det(a) det(ka) = k n det(a) det(a + B) det(a) + det(b) (In some cases this is true but not always) A is invertible

### Midterm Exam I, Calculus III, Sample A

Midterm Exam I, Calculus III, Sample A 1. (1 points) Show that the 4 points P 1 = (,, ), P = (, 3, ), P 3 = (1, 1, 1), P 4 = (1, 4, 1) are coplanar (they lie on the same plane), and find the equation of

### Geometry Unit 1. Basics of Geometry

Geometry Unit 1 Basics of Geometry Using inductive reasoning - Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture- an unproven statement that is based

### Similarity 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

### Vector 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

### AP 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

### Geometry - Chapter 2 Review

Name: Class: Date: Geometry - Chapter 2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine if the conjecture is valid by the Law of Syllogism.

### Numerical 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

Announcements 2-D Vector Addition Today s Objectives Understand the difference between scalars and vectors Resolve a 2-D vector into components Perform vector operations Class Activities Applications Scalar

### MATH 275: Calculus III. Lecture Notes by Angel V. Kumchev

MATH 275: Calculus III Lecture Notes by Angel V. Kumchev Contents Preface.............................................. iii Lecture 1. Three-Dimensional Coordinate Systems..................... 1 Lecture

### Vectors 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

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

### The 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

### Determinants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number

LECTURE 13 Determinants 1. Calculating the Area of a Parallelogram Definition 13.1. Let A be a matrix. [ a c b d ] The determinant of A is the number det A) = ad bc Now consider the parallelogram formed

### MATH 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

### GCE 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

### Chapter 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

### Section 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

### The 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

### Higher Geometry Problems

Higher Geometry Problems ( Look up Eucidean Geometry on Wikipedia, and write down the English translation given of each of the first four postulates of Euclid. Rewrite each postulate as a clear statement

### ex) 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

### Definition: 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.

### L 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:

### = = 4 + = 4 + = 25 = 5

1 4 1 4 1 4 = 4 6+ 1 = 4 + = 4 + = 25 = 5 5. Find the side length of each square as a square root. Then estimate the square root. A B C D A side length B side length C side length D side length = 4 2+

### Modern Geometry Homework.

Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z

### Orthogonal 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).

### ENGR-1100 Introduction to Engineering Analysis. Lecture 3

ENGR-1100 Introduction to Engineering Analysis Lecture 3 POSITION VECTORS & FORCE VECTORS Today s Objectives: Students will be able to : a) Represent a position vector in Cartesian coordinate form, from

### Solutions 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

### REVIEW OVER VECTORS. A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example.

REVIEW OVER VECTORS I. Scalars & Vectors: A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example mass = 5 kg A vector is a quantity that can be described

### Useful 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

### Vectors 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

### Solving 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

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to