# 13.4 THE CROSS PRODUCT

Size: px
Start display at page:

Transcription

1 710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems to show (without trigonometry) that the geometric and algebraic definitions of the dot product are equivalent. Let u = u 1 i + u 2 j + u 3 k and = v1 i + v 2 j + v 3 k be any vectors. Write ( u )geom for the result of the dot product computed geometrically. Substitute u = u 1 i + u 2 j + u 3 k and use Problems to expand ( u ) geom. Substitute for and expand. Then calculate the dot products i i, i j, etc. geometrically. 63. For any vectors and, consider the following function of t: q(t) = ( + t ) ( + t ). (a) Explain why q(t) 0 for all real t. (b) Expand q(t) as a quadratic polynomial in t using the properties on page 702. (c) Using the discriminant of the quadratic, show that, THE CROSS PRODUCT In the previous section we combined two vectors to get a number, the dot product. In this section we see another way of combining two vectors, this time to get a vector, the cross product. Any two vectors in 3-space form a parallelogram. We define the cross product using this parallelogram. The Area of a Parallelogram Consider the parallelogram formed by the vectors and with an angle of θ between them. Then Figure shows Area of parallelogram = Base Height = sin θ. How would we compute the area of the parallelogram if we were given and in components, = v 1 i + v 2 j + v 3 k and = w1 i + w 2 j + w 3 k? Project 1 on page 721 shows that if and are in the xy-plane, so v 3 = w 3 = 0, then Area of parallelogram = v 1 w 2 v 2 w 1. What if and do not lie in the xy-plane? The cross product will enable us to compute the area of the parallelogram formed by any two vectors. sin θ Definition of the Cross Product θ Figure 13.35: Parallelogram formed by and has Area = sin θ We define the cross product of the vectors and, written, to be a vector perpendicular to both and. The magnitude of this vector is the area of the parallelogram formed by the two vectors. The direction of is given by the normal vector, n, to the plane defined by and. If we require that n be a unit vector, there are two choices for n, pointing out of the plane in opposite directions. We pick one by the following rule (see Figure 13.36): The right-hand rule: Place and so that their tails coincide and curl the fingers of your right hand through the smaller of the two angles from to ; your thumb points in the direction of the normal vector, n.

2 Like the dot product, there are two equivalent definitions of the cross product: 13.4 THE CROSS PRODUCT 711 The following two definitions of the cross product or vector product are equivalent: Geometric definition If and are not parallel, then ( ) Area of parallelogram = n = ( sin θ) n, with edges and where 0 θ π is the angle between and and n is the unit vector perpendicular to and pointing in the direction given by the right-hand rule. If and are parallel, then = 0. Algebraic definition = (v 2 w 3 v 3 w 2 ) i + (v 3 w 1 v 1 w 3 ) j + (v 1 w 2 v 2 w 1 ) k where = v 1 i + v 2 j + v 3 k and = w1 i + w 2 j + w 3 k. Notice that the magnitude of the k component is the area of a 2-dimensional parallelogram and the other components have a similar form. Problems 40 and 37 at the end of this section show that the geometric and algebraic definitions of the cross product give the same result. = Area of parallelogram Unit normal determined by right-hand rule θ Figure 13.36: Area of parallelogram = Figure 13.37: The cross product Unlike the dot product, the cross product is only defined for three-dimensional vectors. The geometric definition shows us that the cross product is rotation invariant. Imagine the two vectors and as two metal rods welded together. Attach a third rod whose direction and length correspond to. (See Figure ) Then, no matter how we turn this set of rods, the third will still be the cross product of the first two. The algebraic definition is more easily remembered by writing it as a 3 3 determinant. (See Appendix E.) = v 1 v 2 v 3 = (v 2 w 3 v 3 w 2 ) i + (v 3 w 1 v 1 w 3 ) j + (v 1 w 2 v 2 w 1 ) k. w 1 w 2 w 3 Example 1 Find i j and j i. Solution The vectors i and j both have magnitude 1 and the angle between them is π/2. By the right-hand rule, the vector i j is in the direction of k, so n = k and we have ( i j = i j sin π ) k = k. 2

3 712 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS Similarly, the right-hand rule says that the direction of j i is k, so j i = ( j i sin π 2 )( k ) = k. Similar calculations show that j k = i and k i = j. Example 2 For any vector, find. Solution Since is parallel to itself, = 0. Example 3 Solution Find the cross product of = 2 i + j 2 k and = 3 i + k and check that the cross product is perpendicular to both and. Writing as a determinant and expanding it into three two-by-two determinants, we have 1 2 = = i j k 3 0 = i (1(1) 0( 2)) j (2(1) 3( 2)) + k (2(0) 3(1)) = i 8 j 3 k. To check that is perpendicular to, we compute the dot product: Similarly, ( ) = (2 i + j 2 k ) ( i 8 j 3 k ) = = 0. ( ) = (3 i + 0 j + k ) ( i 8 j 3 k ) = = 0. Thus, is perpendicular to both and. Properties of the Cross Product The right-hand rule tells us that and point in opposite directions. The magnitudes of and are the same, so = ( ). (See Figure ) Figure 13.38: Diagram showing = ( ) This explains the first of the following properties. The other two are derived in Problems 31, 32, and 40 at the end of this section. Properties of the Cross Product For vectors u,, and scalar λ 1. = ( ) 2. (λ ) = λ( ) = (λ ) 3. u ( + ) = u + u.

4 The Equivalence of the Two Definitions of the Cross Product 13.4 THE CROSS PRODUCT 713 Problem 40 on page 716 uses geometric arguments to show that the cross product distributes over addition. Problem 37 then shows how the formula in the algebraic definition of the cross product can be derived from the geometric definition. The Equation of a Plane Through Three Points The equation of a plane is determined by a point P 0 = (x 0,y 0,z 0 ) on the plane, and a normal vector, n = a i + b j + c k: a(x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0. However, a plane can also be determined by three points on it (provided they do not lie on a line). In that case we can find an equation of the plane by first determining two vectors in the plane and then finding a normal vector using the cross product, as in the following example. Example 4 Find an equation of the plane containing the points P = (1,3,0), Q = (3,4, 3), and R = (3,6,2). Solution Since the points P and Q are in the plane, the displacement vector between them, PQ, is in the plane, where PQ = (3 1) i + (4 3) j + ( 3 0) k = 2 i + j 3 k. The displacement vector PR is also in the plane, where PR = (3 1) i + (6 3) j + (2 0) k = 2 i + 3 j + 2 k. Thus, a normal vector, n, to the plane is given by n = PQ PR = = 11 i 10 j + 4 k Since the point (1,3,0) is on the plane, the equation of the plane is which simplifies to 11(x 1) 10(y 3) + 4(z 0) = 0, 11x 10y + 4z = 19. You should check that P, Q, and R satisfy the equation of the plane. Areas and Volumes Using the Cross Product and Determinants We can use the cross product to calculate the area of the parallelogram with sides and. We say that is the area vector of the parallelogram. The geometric definition of the cross product tells us that is normal to the parallelogram and gives us the following result: Area of a parallelogram with edges = v 1 i + v 2 j + v 3 k and = w1 i + w 2 j + w 3 k is given by Area =, where = v 1 v 2 v 3. w 1 w 2 w 3

5 714 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS Example 5 Find the area of the parallelogram with edges = 2 i + j 3 k and = i + 3 j + 2 k. Solution We calculate the cross product: = = (2 + 9) i (4 + 3) j + (6 1) k = 11 i 7 j + 5 k The area of the parallelogram with edges and is the magnitude of the vector : Area = = ( 7) = 195. Volume of a Parallelepiped Consider the parallelepiped with sides formed by,, and. (See Figure ) Since the base is formed by the vectors and, we have Area of base of parallelepiped =. θ θ Figure 13.39: Volume of a Parallelepiped Figure 13.40: The vectors,, are called a right-handed set Figure 13.41: The vectors,, are called a left-handed set The vectors,, and can be arranged either as in Figure or as in Figure In either case, Height of parallelepiped = cos θ, where θ is the angle shown in the figures. In Figure the angle θ is less than π/2, so the product, ( ), called the triple product, is positive. Thus, in this case Volume of parallelepiped = Base Height = cos θ = ( ). In Figure 13.41, the angle, π θ, between and is more than π/2, so the product ( ) is negative. Thus, in this case we have Volume = Base Height = cos θ = cos(π θ) = ( b ) = ( ). Therefore, in both cases the volume is given by ( b ). Using determinants, we can write Volume of a parallelepiped with edges,, is given by Volume = ( ) = Absolute value of the determinant a 1 a 2 a 3 b 1 b 2 b 3. c 1 c 2 c 3

6 13.4 THE CROSS PRODUCT 715 Exercises and Problems for Section 13.4 Exercises In Exercises 1 7, use the algebraic definition to find. 1. = k, = j 2. = i, = j + k 3. = i + k, = i + j 4. = i + j + k, = i + j + k 5. = 2 i 3 j + k, = i + 2 j k 6. = 2 i j k, = 6 i + 3 j + 3 k 7. = 3 i + 5 j + 4 k, = i 3 j k Use the geometric definition in Exercises 8 9 to find: 8. 2 i ( i + j ) 9. ( i + j ) ( i j ) Problems 16. Find a vector parallel to the line of intersection of the planes given by the equations 2x 3y + 5z = 2 and 4x + y 3z = Find the equation of the plane through the origin which is perpendicular to the line of intersection of the planes in Problem Find the equation of the plane through the point (4,5, 6) and perpendicular to the line of intersection of the planes in Problem Find an equation for the plane through the origin containing the points (1, 3, 0) and (2, 4, 1). 20. Find a vector parallel to the line of intersection of the two planes 4x 3y + 2z = 12 and x + 5y z = Find a vector parallel to the intersection of the planes 2x 3y + 5z = 2 and 4x + y 3z = Find the equation of the plane through the origin which is perpendicular to the line of intersection of the planes in Problem Find the equation of the plane through the point (4,5, 6) which is perpendicular to the line of intersection of the planes in Problem Find the equation of a plane through the origin and perpendicular to x y + z = 5 and 2x + y 2z = Let P = (0,1, 0), Q = ( 1, 1,2), R = (2, 1, 1). Find (a) The area of the triangle PQR. (b) The equation for a plane that contains P, Q, and R. 26. Let A = ( 1, 3, 0), B = (3, 2,4), and C = (1, 1, 5). (a) Find an equation for the plane that passes through these three points. (b) Find the area of the triangle determined by these three points. In Exercises 10 11, use the properties on page 712 to find: 10. ( ( i + j ) i ) j 11. ( i + j ) ( i j ) Find an equation for the plane through the points in Exercises (1,0, 0), (0,1, 0), (0,0, 1). 13. (3,4, 2), ( 2, 1,0), (0, 2,1). 14. For = 3 i + j k and = i 4 j +2 k, find and check that it is perpendicular to both and. 15. If = 3 i 2 j + 4 k and = i + 2 j k, find and. What is the relation between the two answers? 27. If and are both parallel to the xy-plane, what can you conclude about? Explain. 28. Suppose = 5 and = 3, and the angle between and is θ. Find (a) tan θ (b) θ. 29. If = 2 i 3 j + 5 k, and = 3, find tan θ where θ is the angle between and. 30. The point P in Figure has position vector obtained by rotating the position vector r of the point (x, y) by 90 counterclockwise about the origin. (a) Use the geometric definition of the cross product to explain why = k r. (b) Find the coordinates of P. P y r Figure (x, y) 31. Use the algebraic definition to check that ( + ) = ( ) + ( ). 32. If and are nonzero vectors, use the geometric definition of the cross product to explain why (λ ) = λ( ) = (λ ). Consider the cases λ > 0, and λ = 0, and λ < 0 separately. x

7 716 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 33. Use a parallelepiped to show that ( ) = ( ) for any vectors,, and. 34. Show that 2 = 2 2 ( b ) If + + = 0, show that outward pointing area vectors of the faces equals the zero vector. = =. Geometrically, what does this imply about,, and? 36. If = a 1 i + a 2 j + a 3 k, = b1 i + b 2 j + b 3 k and = c 1 i + c 2 j + c 3 k are any three vectors in space, show that a 1 a 2 a 3 ( ) = b 1 b 2 b 3. c 1 c 2 c Use the fact that i i = 0, i j = k, i k = j, and so on, together with the properties on page 712 to derive the algebraic definition for the cross product. 38. In this problem, we arrive at the algebraic definition for the cross product by a different route. Let = a 1 i + a 2 j +a 3 k and = b1 i +b 2 j +b 3 k. We seek a vector = x i + y j + z k which is perpendicular to both and. Use this requirement to construct two equations for x, y, and z. Eliminate x and solve for y in terms of z. Then eliminate y and solve for x in terms of z. Since z can be any value whatsoever (the direction of is unaffected), select the value for z which eliminates the denominator in the equation you obtained. How does the resulting expression for compare to the formula we derived on page 711? 39. For vectors and, let = ( ). (a) Show that lies in the plane containing and. (b) Use Problems 33 and 34 to show that = 0 and = 2 2 ( b ) 2. (c) Show that ( ) = 2 ( ). Figure In Problems 42 44, find the vector representing the area of a surface. The magnitude of the vector equals the magnitude of the area; the direction is perpendicular to the surface. Since there are two perpendicular directions, we pick one by giving an orientation for the surface. 42. The rectangle with vertices (0, 0,0), (0, 1,0), (2, 1, 0), and (2, 0, 0), oriented so that it faces downward. 43. The circle of radius 2 in the yz-plane, facing in the direction of the positive x-axis. 44. The triangle ABC, oriented upward, where A = (1, 2, 3), B = (3, 1,2), and C = (2,1, 3). 45. This problem relates the area of a parallelogram S lying in the plane z = mx+ny+c to the area of its projection R in the xy-plane. Let S be determined by the vectors u = u 1 i +u 2 j +u 3 k and = v1 i +v 2 j +v 3 k. See Figure (a) Find the area of S (b) Find the area of R (c) Find m and n in terms of the components of u and. (d) Show that Area of S = 1 + m 2 + n 2 Area of R 40. Use the result of Problem 33 to show that the cross product distributes over addition. First, use distributivity for the dot product to show that for any vector d, [( + ) ] d = [( ) + ( )] d. Next, show that for any vector d, z u S [(( + ) ) ( ) ( )] d = 0. y Finally, explain why you can conclude that ( + ) = ( ) + ( ). 41. Figure shows the tetrahedron determined by three vectors,,. The area vector of a face is a vector perpendicular to the face, pointing outward, whose magnitude is the area of the face. Show that the sum of the four x R Figure 13.44

8 REVIEW EXERCISES AND PROBLEMS FOR CHAPTER THIRTEEN 717 CHAPTER SUMMARY (see also Ready Reference at the end of the book) Vectors Geometric definition of vector addition, subtraction and scalar multiplication, resolving into i, j, and k components, magnitude of a vector, algebraic properties of addition and scalar multiplication. Dot Product Geometric and algebraic definition, algebraic properties, using dot products to find angles and determine perpendicularity, the equation of a plane with given normal vector passing through a given point, projection of a vector in a direction given by a unit vector. Cross Product Geometric and algebraic definition, algebraic properties, cross product and volume, finding the equation of a plane through three points. REVIEW EXERCISES AND PROBLEMS FOR CHAPTER THIRTEEN Exercises In Exercises 1 2, is the quantity a vector or a scalar? Compute it. 1. u, where u = 2 i 3 j 4 k and = k j 2. u, where u = 2 i 3 j 4 k and = 3 i j + k. 3. Resolve the vectors in Figure into components. For Exercises 5 7, perform the indicated operations on the following vectors: = i + 6 j, x = 2 i + 9 j, y = 4 i 7 j x + y 7. x y 5 4 In Exercises 8 17, use = 2 i +3 j k and = i j +2 k to calculate the given quantities d e f Figure x ( ) 15. ( ) 16. ( ) 17. ( ) ( ) 4. Resolve vector into components if = 8 and the direction of is shown in Figure In Exercises 18 19, find a normal vector to the plane. y 40 Figure x 18. 2x + y z = (x z) = 3(x + y) 20. Find the equation of the plane through the origin which is parallel to z = 4x 3y Let = 3 i + 2 j 2 k and = 4 i 3 j + k. Find each of the following: (a) (b) (c) A vector of length 5 parallel to vector (d) The angle between vectors and (e) The component of in the direction of (f) A vector perpendicular to vector (g) A vector perpendicular to both vectors and

### 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

### 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)

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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 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

### 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

### 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

### 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,

### 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

### 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

### 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,

### 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

### 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

### 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

### 1.5 Equations of Lines and Planes in 3-D

40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3-D Recall that given a point P = (a, b, c), one can draw a vector from

### Review Sheet for Test 1

Review Sheet for Test 1 Math 261-00 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

### Math 215 HW #6 Solutions

Math 5 HW #6 Solutions Problem 34 Show that x y is orthogonal to x + y if and only if x = y Proof First, suppose x y is orthogonal to x + y Then since x, y = y, x In other words, = x y, x + y = (x y) T

### 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

### 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,

### 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:

### 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.

### 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

### ... ... . (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,

### 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

### 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,

### 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

### 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

### 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

### Section 11.1: Vectors in the Plane. Suggested Problems: 1, 5, 9, 17, 23, 25-37, 40, 42, 44, 45, 47, 50

Section 11.1: Vectors in the Plane Page 779 Suggested Problems: 1, 5, 9, 17, 3, 5-37, 40, 4, 44, 45, 47, 50 Determine whether the following vectors a and b are perpendicular. 5) a = 6, 0, b = 0, 7 Recall

### Vector Algebra. Addition: (A + B) + C = A + (B + C) (associative) Subtraction: A B = A + (-B)

Vector Algebra When dealing with scalars, the usual math operations (+, -, ) are sufficient to obtain any information needed. When dealing with ectors, the magnitudes can be operated on as scalars, but

### 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.

### Solutions for Review Problems

olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

### 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

### 1. Vectors and Matrices

E. 8.02 Exercises. Vectors and Matrices A. Vectors Definition. A direction is just a unit vector. The direction of A is defined by dir A = A, (A 0); A it is the unit vector lying along A and pointed like

### 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

### 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,

### 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

### 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

### Eðlisfræði 2, vor 2007

[ Assignment View ] [ Pri Eðlisfræði 2, vor 2007 28. Sources of Magnetic Field Assignment is due at 2:00am on Wednesday, March 7, 2007 Credit for problems submitted late will decrease to 0% after the deadline

### a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a

Bellwork a.) Write the line 2x - 4y = 9 into slope intercept form b.) Find the slope of the line parallel to part a c.) Find the slope of the line perpendicular to part b or a May 8 7:30 AM 1 Day 1 I.

### Concepts in Calculus III

Concepts in Calculus III Beta Version UNIVERSITY PRESS OF FLORIDA Florida A&M University, Tallahassee Florida Atlantic University, Boca Raton Florida Gulf Coast University, Ft. Myers Florida International

### LINES AND PLANES CHRIS JOHNSON

LINES AND PLANES CHRIS JOHNSON Abstract. In this lecture we derive the equations for lines and planes living in 3-space, as well as define the angle between two non-parallel planes, and determine the distance

### 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

### Solutions to Exercises, Section 5.1

Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle

### 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

### 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

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### CHAPTER FIVE. 5. Equations of Lines in R 3

118 CHAPTER FIVE 5. Equations of Lines in R 3 In this chapter it is going to be very important to distinguish clearly between points and vectors. Frequently in the past the distinction has only been a

### x1 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

### 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

### 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

### 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

### 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

### Chapter 22: Electric Flux and Gauss s Law

22.1 ntroduction We have seen in chapter 21 that determining the electric field of a continuous charge distribution can become very complicated for some charge distributions. t would be desirable if we

### 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

### Chapter 11 Equilibrium

11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of

### 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,

### Section 2.4: Equations of Lines and Planes

Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y

### Equations of Lines and Planes

Calculus 3 Lia Vas Equations of Lines and Planes Planes. A plane is uniquely determined by a point in it and a vector perpendicular to it. An equation of the plane passing the point (x 0, y 0, z 0 ) perpendicular

### 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,

### Exam 1 Sample Question SOLUTIONS. y = 2x

Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can

### 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

### PHYSICS 151 Notes for Online Lecture #6

PHYSICS 151 Notes for Online Lecture #6 Vectors - A vector is basically an arrow. The length of the arrow represents the magnitude (value) and the arrow points in the direction. Many different quantities

### 10.5. Click here for answers. Click here for solutions. EQUATIONS OF LINES AND PLANES. 3x 4y 6z 9 4, 2, 5. x y z. z 2. x 2. y 1.

SECTION EQUATIONS OF LINES AND PLANES 1 EQUATIONS OF LINES AND PLANES A Click here for answers. S Click here for solutions. 1 Find a vector equation and parametric equations for the line passing through

### 3. KINEMATICS IN TWO DIMENSIONS; VECTORS.

3. KINEMATICS IN TWO DIMENSIONS; VECTORS. Key words: Motion in Two Dimensions, Scalars, Vectors, Addition of Vectors by Graphical Methods, Tail to Tip Method, Parallelogram Method, Negative Vector, Vector

### Linear 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

### 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

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### 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

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### 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

### Chapter 3 Vectors. m = m1 + m2 = 3 kg + 4 kg = 7 kg (3.1)

COROLLARY I. A body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately. Isaac Newton - Principia

### Analysis of Stresses and Strains

Chapter 7 Analysis of Stresses and Strains 7.1 Introduction axial load = P / A torsional load in circular shaft = T / I p bending moment and shear force in beam = M y / I = V Q / I b in this chapter, we

### 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

Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are

### Mechanics 1: Conservation of Energy and Momentum

Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

### POINT OF INTERSECTION OF TWO STRAIGHT LINES

POINT OF INTERSECTION OF TWO STRAIGHT LINES THEOREM The point of intersection of the two non parallel lines bc bc ca ca a x + b y + c = 0, a x + b y + c = 0 is,. ab ab ab ab Proof: The lines are not parallel

### Math Placement Test Practice Problems

Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211