# 9.4. The Scalar Product. Introduction. Prerequisites. Learning Style. Learning Outcomes

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 two vectors is calculated the result is a scalar. The second product is known as the vector product. When this is calculated the result is a vector. The definitions of these products may seem rather strange at first, but they are widely used in applications. In this Block we consider only the scalar product. Prerequisites Before starting this Block you should... Learning Outcomes After completing this Block you should be able to... calculate, from its definition, the scalar product of two given vectors calculate the scalar product of two vectors given in cartesian form use the scalar product to find the angle between two vectors use the scalar product to test whether two vectors are perpendicular 1 that a vector can be represented as a directed line segment 2 how to express a vector in cartesian form 3 how to find the modulus of a vector Learning Style To achieve what is expected of you... allocate sufficient study time briefly revise the prerequisite material attempt every guided exercise and most of the other exercises

2 1. Definition of the Scalar Product Consider the two vectors a and b shown in Figure??. b θ a Figure 1: Two vectors subtend an angle θ. Note that the tails of the two vectors coincide and that the angle between the vectors has been labelled θ. Their scalar product, denoted by a b, is defined as the product a b cos θ. It is very important to use the dot in the formula. The dot is the specific symbol for the scalar product, and is the reason why the scalar product is also known as the dot product. You should not use a sign in this context because this sign is reserved for the vector product which is quite different. The angle θ is always chosen to lie between 0 and π, and the tails of the two vectors should coincide. Figure?? shows two incorrect ways of measuring θ. b b θ a a θ Figure 2: θ should not be measured in these ways. We can remember this formula as: scalar product : a b = a b cos θ the modulus of the first vector, multiplied by the modulus of the second vector, multiplied by the cosine of the angle between them. Clearly b a = b a cos θ and so a b = b a. Thus we can evaluate a scalar product in any order: the operation is said to be commutative. Engineering Mathematics: Open Learning Unit Level 1 2

3 Example Vectors a and b are shown in the figure below. Vector a has modulus 6 and vector b has modulus 7 and the angle between them is 60. Calculate a.b. b 60 a The angle between the two vectors is 60. Hence a b = a b cos θ = (6)(7) cos 60 =21 The scalar product of a and b is equal to 21. Note that when finding a scalar product the result is always a scalar. Example Find i i where i is the unit vector in the direction of the positive x axis. Because i is a unit vector its modulus is 1. Also, the angle between i and itself is zero. Therefore i.i = (1)(1) cos 0 =1 So the scalar product of i with itself equals 1. It is easy to verify that j.j = 1 and k.k =1. Example Find i j where i and j are unit vectors in the directions of the x and y axes. Because i and j are unit vectors they each have a modulus of 1. The angle between the two vectors is 90. Therefore i j = (1)(1) cos 90 =0 That is i j =0. More generally, the following results are easily verified: i i = j j = k k =1 i j = i k = j k =0 3 Engineering Mathematics: Open Learning Unit Level 1

4 Even more generally, whenever any two vectors are perpendicular to each other their scalar product is zero because the angle between the vectors is 90 and cos 90 =0. The scalar product of perpendicular vectors is zero. 2. A Formula for Finding the Scalar Product We can use the previous results to obtain a formula for finding a scalar product when the vectors are given in cartesian form. We consider vectors in the xy plane. Suppose a = a 1 i + a 2 j and b = b 1 i + b 2 j. Then a b = (a 1 i + a 2 j) (b 1 i + b 2 j) = a 1 i (b 1 i + b 2 j)+a 2 j (b 1 i + b 2 j) = a 1 b 1 i i + a 1 b 2 i j + a 2 b 1 j i + a 2 b 2 j j Now, using the previous boxed results we can simplify this to give the following formula: if a = a 1 i + a 2 j and b = b 1 i + b 2 j then a b = a 1 b 1 + a 2 b 2 Thus to find the scalar product of two vectors their i components are multiplied together, their j components are multiplied together and the results are added. Example If a =7i +8j and b =5i 2j, find the scalar product a b. We use the previous boxed formula and multiply corresponding components together, adding the results. a b = (7i +8j) (5i 2j) = (7)(5) + (8)( 2) = = 19 The formula readily generalises to vectors in three dimensions as follows: if a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k then a b = a 1 b 1 + a 2 b 2 + a 3 b 3 Engineering Mathematics: Open Learning Unit Level 1 4

5 Example If a =5i +3j 2k and b =8i 9j +11k, find a b. Corresponding components are multiplied together and the results are added. a b = (5)(8) + (3)( 9)+( 2)(11) = = 9 Note again that the result is a scalar: there are no i s, j s, or k s in the answer. Now do this exercise If p =4i 3j +7k and q =6i j +2k, find p q. Corresponding components are multiplied together and the results are added. Now do this exercise If r =3i +2j +9k find r r. Show that this is the same as r 2. Answer Answer For any vector r we have r 2 = r r 3. Resolving One Vector Along Another The scalar product can be used to find the component of a vector in the direction of another vector. Consider Figure?? which shows two arbitrary vectors a and n. Let ˆn be a unit vector in the direction of n. P a O ˆn θ Q n projection of a onto n Figure 3: 5 Engineering Mathematics: Open Learning Unit Level 1

6 Study the figure carefully and note that a perpendicular has been drawn from P to meet n at Q. The distance OQ is called the projection of a onto n. Simple trigonometry tells us that the length of the projection is a cos θ. Now by taking the scalar product of a with the unit vector ˆn we find a ˆn = a ˆn cos θ = a cos θ since ˆn =1 We conclude that a ˆn is the component of a in the direction of n Example The figure below shows a plane containing the point A with position vector a. The vector ˆn is a unit vector perpendicular to the plane (such a vector is called a normal vector). Find an expression for the perpendicular distance of the plane from the origin. ˆn ˆn l A O a From the diagram note that the perpendicular distance l of the plane from the origin is the projection of a onto ˆn and is thus a ˆn. 4. Using the Scalar Product to Find the Angle Between Two Vectors We have two distinct ways of calculating the scalar product of two vectors. From the first Key Point of this block a b = a b cos θ whislst from the last of Section 2 we have a b = a 1 b 1 + a 2 b 2 + a 3 b 3. Both methods of calculating the scalar product are entirely equivalent and will always give the same value for the scalar product. We can exploit this correspondence to find the angle between two vectors. The following example illustrates the procedure to be followed. Engineering Mathematics: Open Learning Unit Level 1 6

7 Example Find the angle between the vectors a =5i +3j 2k and b =8i 9j +11k. The scalar product of these two vectors has already been found in the example on page 4 to be 9. The modulus of a is ( 2) 2 = 38. The modulus of b is 8 2 +( 9) = 266. Substituting these into the formula for the scalar product we find a b = a b cos θ 9 = cos θ from which so that cos θ = = θ = cos 1 ( ) = In general, the angle between two vectors can be found from the following formula: The angle θ between vectors a, b is such that: cos θ = a b a b More exercises for you to try 1. If a =2i 5j and b =3i +2j find a b and verify that a b = b a. 2. Find the angle between p =3i j and q = 4i +6j. 3. Use the definition of the scalar product to show that if two vectors are perpendicular, their scalar product is zero. 4. If a and b are perpendicular, simplify (a 2b) (3a +5b). 5. If p = i +8j +7k and q =3i 2j +5k, find p q. 6. Show that the vectors 1 i + j and 2i j are perpendicular The work done by a force F in moving a body through a displacement r is given by F r. Find the work done by the force F =3i +7k if it causes a body to move from the point with coordinates (1, 1, 2) to the point (7, 3, 5). 8. Find the angle between the vectors i j k and 2i + j +2k. Answer 7 Engineering Mathematics: Open Learning Unit Level 1

8 5. Computer Exercise or Activity For this exercise it will be necessary for you to access the computer package DERIVE. DERIVE has routines for dealing with two- and threedimensional vectors. To use vectors you can enter them by keying Author:Vector. When requested for the dimension respond with either 2 or 3 as appropriate. You will then be able to enter your vectors. Alternatively, and this is probably easier, use Author:Expression and enclose your vector within square brackets. For example the vector 2i +3j 4k would be entered as [2, 3, 4]. The product of a scalar with a vector is written in an obvious way. The vector 6(2i +3j 4k) would be keyed in as Author:Expression 6[2, 3, 4]. DERIVE responds with [12, 18, 24]. The modulus of a vector is obtained using the abs command. For example to find 2i+3j 4k you would key in Author:Expression abs[2, 3, 4]. DERIVE responds with 29. The scalar product can be obtained in DERIVE by using the. notation. To obtain (2i +3j 4k).( 1i +3j +5k) you would key in Author:Expression [2, 3, 4].[ 1, 3, 5]. DERIVE responds with 13. As a useful exercise use DERIVE to obtain the solutions to Exercises 1,5,6 and 8. Engineering Mathematics: Open Learning Unit Level 1 8

9 End of Block Engineering Mathematics: Open Learning Unit Level 1

10 41 Back to the theory Engineering Mathematics: Open Learning Unit Level 1 10

11 94. r = = 94 hence r 2 = r r. Back to the theory 11 Engineering Mathematics: Open Learning Unit Level 1

12 , 4. 3a 2 10b units Back to the theory Engineering Mathematics: Open Learning Unit Level 1 12

### 9.3. Direction Ratios and Direction Cosines. Introduction. Prerequisites. Learning Outcomes. Learning Style

Direction Ratios and Direction Cosines 9.3 Introduction Direction ratios provide a convenient way of specifying the direction of a line in three dimensional space. Direction cosines are the cosines of

### 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

### 9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

### Section V.3: Dot Product

Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,

### 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

### 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,

### 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

### 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

### 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

### 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

### 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

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

### 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.

### 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

### 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

### 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

### 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 >

### Vector 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

### 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

### 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

### Section 9.5: Equations of Lines and Planes

Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 3-5 odd, 2-37 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that

### 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

### 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

### 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

### 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

### 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

### 2 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

### 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

### Vectors SCALAR PRODUCT. Graham S McDonald. A Tutorial Module for learning about the scalar product of two vectors. Table of contents Begin Tutorial

Vectors SCALAR PRODUCT Graham S McDonald A Tutorial Module for learning about the scalar product of two vectors Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk 1. Theory 2. Exercises

### 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,

### 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,

### 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

### 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

### 3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

### 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

### 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

### 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

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Equilibrium of Concurrent Forces (Force Table)

Equilibrium of Concurrent Forces (Force Table) Objectives: Experimental objective Students will verify the conditions required (zero net force) for a system to be in equilibrium under the influence of

### 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

### 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

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

### 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

### 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

### 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

### 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

### 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

### Mathematics 205 HWK 6 Solutions Section 13.3 p627. Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors.

Mathematics 205 HWK 6 Solutions Section 13.3 p627 Note: Remember that boldface is being used here, rather than overhead arrows, to indicate vectors. Problem 5, 13.3, p627. Given a = 2j + k or a = (0,2,

### discuss 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

### JUST THE MATHS UNIT NUMBER 8.5. VECTORS 5 (Vector equations of straight lines) A.J.Hobson

JUST THE MATHS UNIT NUMBER 8.5 VECTORS 5 (Vector equations of straight lines) by A.J.Hobson 8.5.1 Introduction 8.5. The straight line passing through a given point and parallel to a given vector 8.5.3

### Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

### Applications of Trigonometry

chapter 6 Tides on a Florida beach follow a periodic pattern modeled by trigonometric functions. Applications of Trigonometry This chapter focuses on applications of the trigonometry that was introduced

### Mechanics lecture 7 Moment of a force, torque, equilibrium of a body

G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and

### 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.

### 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

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

### Solving simultaneous equations using the inverse matrix

Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix

### LINES AND PLANES IN R 3

LINES AND PLANES IN R 3 In this handout we will summarize the properties of the dot product and cross product and use them to present arious descriptions of lines and planes in three dimensional space.

### 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:

### Lab: Vectors. You are required to finish this section before coming to the lab. It will be checked by one of the lab instructors when the lab begins.

Lab: Vectors Lab Section (circle): Day: Monday Tuesday Time: 8:00 9:30 1:10 2:40 Name Partners Pre-Lab You are required to finish this section before coming to the lab. It will be checked by one of the

### ectors and Application P(x, y, z)! \$ ! \$ & " 11,750 12,750 13,750

thstrack MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) odule 3 Topic 3 Module 9 Introduction Vectors and Applications to Matrices ectors

### Section 13.5 Equations of Lines and Planes

Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.

### 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

### Addition and Subtraction of Vectors

ddition and Subtraction of Vectors 1 ppendi ddition and Subtraction of Vectors In this appendi the basic elements of vector algebra are eplored. Vectors are treated as geometric entities represented b

### Recall 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

### 10.4. De Moivre s Theorem. Introduction. Prerequisites. Learning Outcomes. Learning Style

De Moivre s Theorem 10.4 Introduction In this block we introduce De Moivre s theorem and examine some of its consequences. We shall see that one of its uses is in obtaining relationships between trigonometric

### Physics 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

### Vectors. Physics 115: Spring Introduction

Vectors Physics 115: Spring 2005 Introduction The purpose of this laboratory is to examine the hypothesis that forces are vectors and obey the law of vector addition and to observe that the vector sum

### Basic Concepts of Crystallography

Basic Concepts of Crystallography Language of Crystallography: Real Space Combination of local (point) symmetry elements, which include angular rotation, center-symmetric inversion, and reflection in mirror

### Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing.

Figuring out the amplitude of the sun an explanation of what the spreadsheet is doing. The amplitude of the sun (at sunrise, say) is the direction you look to see the sun come up. If it s rising exactly

### Lesson 19: Equations for Tangent Lines to Circles

Student Outcomes Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### How is a vector rotated?

How is a vector rotated? V. Balakrishnan Department of Physics, Indian Institute of Technology, Madras 600 036 Appeared in Resonance, Vol. 4, No. 10, pp. 61-68 (1999) Introduction In an earlier series

### 1.3 Displacement in Two Dimensions

1.3 Displacement in Two Dimensions So far, you have learned about motion in one dimension. This is adequate for learning basic principles of kinematics, but it is not enough to describe the motions of

### Linear Equations and Inverse Matrices

Chapter 4 Linear Equations and Inverse Matrices 4. Two Pictures of Linear Equations The central problem of linear algebra is to solve a system of equations. Those equations are linear, which means that

### Section 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,

### The Cartesian Plane The Cartesian Plane. Performance Criteria 3. Pre-Test 5. Coordinates 7. Graphs of linear functions 9. The gradient of a line 13

6 The Cartesian Plane The Cartesian Plane Performance Criteria 3 Pre-Test 5 Coordinates 7 Graphs of linear functions 9 The gradient of a line 13 Linear equations 19 Empirical Data 24 Lines of best fit

### Lab 2: Vector Analysis

Lab 2: Vector Analysis Objectives: to practice using graphical and analytical methods to add vectors in two dimensions Equipment: Meter stick Ruler Protractor Force table Ring Pulleys with attachments

### EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1 - LOADING SYSTEMS TUTORIAL 1 NON-CONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects

### BALTIC OLYMPIAD IN INFORMATICS Stockholm, April 18-22, 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

### 15.3. Calculating centres of mass. Introduction. Prerequisites. Learning Outcomes. Learning Style

Calculating centres of mass 15.3 Introduction In this block we show how the idea of integration as the limit of a sum can be used to find the centre of mass of an object such as a thin plate, like a sheet

### FORCE VECTORS. Lecture s Objectives

CHAPTER Engineering Mechanics: Statics FORCE VECTORS Tenth Edition College of Engineering Department of Mechanical Engineering 2b by Dr. Ibrahim A. Assakkaf SPRING 2007 ENES 110 Statics Department of Mechanical

### 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

### Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col

### 1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

### Two 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

### Trigonometric 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

### The Gradient and Level Sets

The Gradient and Level Sets. Let f(x, y) = x + y. (a) Find the gradient f. Solution. f(x, y) = x, y. (b) Pick your favorite positive number k, and let C be the curve f(x, y) = k. Draw the curve on the

### 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

### 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

### Torgerson s Classical MDS derivation: 1: Determining Coordinates from Euclidean Distances

Torgerson s Classical MDS derivation: 1: Determining Coordinates from Euclidean Distances It is possible to construct a matrix X of Cartesian coordinates of points in Euclidean space when we know the Euclidean

### BCM 6200 - Protein crystallography - I. Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS

BCM 6200 - Protein crystallography - I Crystal symmetry X-ray diffraction Protein crystallization X-ray sources SAXS Elastic X-ray Scattering From classical electrodynamics, the electric field of the electromagnetic

### ANALYTICAL METHODS FOR ENGINEERS

UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

### Units and Vectors: Tools for Physics

Chapter 1 Units and Vectors: Tools for Physics 1.1 The Important Stuff 1.1.1 The SI System Physics is based on measurement. Measurements are made by comparisons to well defined standards which define the

### 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

### 14.1. Basic Concepts of Integration. Introduction. Prerequisites. Learning Outcomes. Learning Style

Basic Concepts of Integration 14.1 Introduction When a function f(x) is known we can differentiate it to obtain its derivative df. The reverse dx process is to obtain the function f(x) from knowledge of

### COMPLEX NUMBERS. -2+2i

COMPLEX NUMBERS Cartesian Form of Complex Numbers The fundamental complex number is i, a number whose square is 1; that is, i is defined as a number satisfying i 1. The complex number system is all numbers

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function