Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS Contents 13.1 Displacement Vectors . . . . . . . . . . . . 686 Notation and Terminology . . . . . . 686 Addition and Subtraction of Displacement Vectors . . . . . . . . . . . . . . . 686 Scalar Multiplication of Displacement Vectors 687 Parallel Vectors . . . . . . . . . . . 688 Components of Displacement Vectors: The Vectors ~i , ~j , and ~ k . . . . . . . . 688 Unit Vectors . . . . . . . . . . . . . . . . 692 13.2 Vectors in General . . . . . . . . . . . . . 694 Velocity Versus Speed . . . . . . . . 694 Acceleration . . . . . . . . . . . . 696 Force . . . . . . . . . . . . . . . 696 Properties of Addition and Scalar Multiplication697 Using Components . . . . . . . . . . . . . 697 Vectors in n Dimensions . . . . . . . . . . 698 Why Do We Want Vectors in n Dimensions? . . . . . . . 698 13.3 The Dot Product . . . . . . . . . . . . . . 701 Definition of the Dot Product . . . . . . . . 701 Why the Two Definitions of the Dot Product Give the Same Result . . . . . . . . 702 Properties of the Dot Product . . . . . . . . 702 Perpendicularity, Magnitude, and Dot Products . . . . . . . . . 703 Using the Dot Product . . . . . . . . . . . 703 Normal Vectors and the Equation of a Plane . . . . . . . . . . . 704 The Dot Product in n Dimensions . . 705 Resolving a Vector into Components: Projections . . . . . . . . . . . . . 705 A Physical Interpretation of the Dot Product: Work . . . . . . 706 13.4 The Cross Product . . . . . . . . . . . . . 710 The Area of a Parallelogram . . . . . . . . 710 Definition of the Cross Product . . . . . . . 710 Properties of the Cross Product . . . . . . . 712 The Equivalence of the Two Definitions of the Cross Product . . . . . . . . . . . 713 The Equation of a Plane Through Three Points 713 Areas and Volumes Using the Cross Product and Determinants . . . . . . . . . . 713 Volume of a Parallelepiped . . . . . 714 REVIEW PROBLEMS . . . . . . . . . . . 717 CHECK YOUR UNDERSTANDING . . . . 720 PROJECTS . . . . . . . . . . . . . . . . . 721 686 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 13.1 DISPLACEMENT VECTORS Suppose you are a pilot planning a flight from Dallas to Pittsburgh. There are two things you must know: the distance to be traveled (so you have enough fuel to make it) and in what direction to go (so you don't miss Pittsburgh). Both these quantities together specify the displacement or displacement vector between the two cities. The displacement vector from one point to another is an arrow with its tail at the first point and its tip at the second. The magnitude (or length) of the displacement vector is the distance between the points, and is represented by the length of the arrow. The direction of the displacement vector is the direction of the arrow. Figure 13.1 shows a map with the displacement vectors from Dallas to Pittsburgh, from Albuquerque to Oshkosh, and from Los Angeles to Buffalo, SD. These displacement vectors have the same length and the same direction. We say that the displacement vectors between the corresponding cities are the same, even though they do not coincide. In other words Displacement vectors which point in the same direction and have the same magnitude are considered to be the same, even if they do not coincide. Buffalo, SD Oshkosh Pittsburgh Los Angeles Albuquerque Dallas Figure 13.1: Displacement vectors between cities Notation and Terminology The displacement vector is our first example of a vector. Vectors have both magnitude and direction; in comparison, a quantity specified only by a number, but no direction, is called a scalar.1 For instance, the time taken by the flight from Dallas to Pittsburgh is a scalar quantity. Displacement is a vector since it requires both distance and direction to specify it. In this book, vectors are written with an arrow over them, ~v , to distinguish them from scalars. Other books use a bold v to denote a vector. We use the notation P Q to denote the displacement vector from a point P to a point Q. The magnitude, or length, of a vector ~v is written k~v k. Addition and Subtraction of Displacement Vectors Suppose NASA commands a robot on Mars to move 75 meters in one direction and then 50 meters in another direction. (See Figure 13.2.) Where does the robot end up? Suppose the displacements are represented by the vectors ~v and w ~ , respectively. Then the sum ~v + w ~ gives the final position. 1 So named by W. R. Hamilton because they are merely numbers on the scale from to . 687 13.1 DISPLACEMENT VECTORS The sum, ~v + w ~ , of two vectors ~v and w ~ is the combined displacement resulting from first applying ~v and then w ~ . (See Figure 13.3.) The sum w ~ + ~v gives the same displacement. Finish Finish ~v Combined displacement w ~ ~v + w ~ 50 m w ~ Start 75 m ~v Start Figure 13.2: Sum of displacements of robots on Mars Figure 13.3: The sum ~v + w ~ =w ~ + ~v Suppose two different robots start from the same location. One moves along a displacement vector ~v and the second along a displacement vector w ~ . What is the displacement vector, ~x , from the first robot to the second? (See Figure 13.4.) Since ~v + ~x = w ~ , we define ~x to be the difference ~x = w ~ ~v . In other words, w ~ ~v gets you from ~v to w ~. The difference, w ~ ~v , is the displacement vector which, when added to ~v , gives w ~ . That is, w ~ = ~v + (w ~ ~v ). (See Figure 13.4.) Second robot ~x = w ~ ~v w ~ First robot ~v Start Figure 13.4: The difference w ~ ~v If the robot ends up where it started, then its total displacement vector is the zero vector, ~0 . The zero vector has no direction. The zero vector, ~0 , is a displacement vector with zero length. Scalar Multiplication of Displacement Vectors If ~v represents a displacement vector, the vector 2~v represents a displacement of twice the magnitude in the same direction as ~v . Similarly, 2~v represents a displacement of twice the magnitude in the opposite direction. (See Figure 13.5.) ~v 2~v 0.5~v 2~v Figure 13.5: Scalar multiples of the vector ~v 688 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS If is a scalar and ~v is a displacement vector, the scalar multiple of ~v by , written ~v , is the displacement vector with the following properties: The displacement vector ~v is parallel to ~v , pointing in the same direction if > 0, and in the opposite direction if < 0. The magnitude of ~v is || times the magnitude of ~v , that is, k~v k = || k~v k . Note that || represents the absolute value of the scalar while k~v k represents the magnitude of the vector ~v . Example 1 Explain why w ~ ~v = w ~ + (1)~v . Solution The vector (1)~v has the same magnitude as ~v , but points in the opposite direction. Figure 13.6 shows that the combined displacement w ~ + (1)~v is the same as the displacement w ~ ~v . (1)~v Finish w ~ ~v w ~ w ~ Start ~v Figure 13.6: Explanation for why w ~ ~v = w ~ + (1)~v Parallel Vectors Two vectors ~v and w ~ are parallel if one is a scalar multiple of the other, that is, if w ~ = ~v , for some scalar . Components of Displacement Vectors: The Vectors i , j , and k Suppose that you live in a city with equally spaced streets running east-west and north-south and that you want to tell someone how to get from one place to another. You'd be likely to tell them how many blocks east-west and how many blocks north-south to go. For example, to get from P to Q in Figure 13.7, we go 4 blocks east and 1 block south. If ~i and ~j are as shown in Figure 13.7, then the displacement vector from P to Q is 4~i ~j . P 4~i ~j Q ~j ~i Figure 13.7: The displacement vector from P to Q is 4~i ~j 13.1 DISPLACEMENT VECTORS 689 We extend the same idea to 3-dimensions. First we choose a Cartesian system of coordinate axes. The three vectors of length 1 shown in Figure 13.8 are the vector ~i , which points along the positive x-axis, the vector ~j , along the positive y-axis, and the vector ~k , along the positive z-axis. z 1 y ~k ~i (3, 2) 2 ~v 1 ~j 3~i 1 1 y x 2~j Figure 13.8: The vectors ~i , ~j and ~k in 3-space 1 2 3 x Figure 13.9: We resolve ~v into components by writing ~v = 3~i + 2~j Writing Displacement Vectors Using i , j , k Any displacement in 3-space or the plane can be expressed as a combination of displacements in the coordinate directions. For example, Figure 13.9 shows that the displacement vector ~v from the origin to the point (3, 2) can be written as a sum of displacement vectors along the x and y-axes: ~v = 3~i + 2~j . This is called resolving ~v into components. In general: We resolve ~v into components by writing ~v in the form ~v = v1~i + v2~j + v3~k . We call v1~i , v2~j , and v3~k the components of ~v . An Alternative Notation for Vectors Many people write a vector in 3-dimensions as a string of three numbers, that is, as ~v = (v1 , v2 , v3 ) instead of ~v = v1~i + v2~j + v3~k . Since the first notation can be confused with a point and the second cannot, we usually use the second form. Example 2 Resolve the displacement vector, ~v , from the point P1 = (2, 4, 10) to the point P2 = (3, 7, 6) into components. Solution To get from P1 to P2 , we move 1 unit in the positive x-direction, 3 units in the positive y-direction, and 4 units in the negative z-direction. Hence ~v = ~i + 3~j 4~k . Example 3 Decide whether the vector ~v = 2~i + 3~j + 5~k is parallel to each of the following vectors: w ~ = 4~i + 6~j + 10~k , ~a = ~i 1.5~j 2.5~k , ~b = 4~i + 6~j + 9~k . Solution Since w ~ = 2~v and ~a = 0.5~v , the vectors ~v , w ~ , and ~a are parallel. However, ~b is not a multiple of ~v (since, for example, 4/2 6= 9/5), so ~v and ~b are not parallel. Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS In general, Figure 13.10 shows us how to express the displacement vector between two points in components: Components of Displacement Vectors The displacement vector from the point P1 = (x1 , y1 , z1 ) to the point P2 = (x2 , y2 , z2 ) is given in components by P1 P2 = (x2 x1 )~i + (y2 y1 )~j + (z2 z1 )~k . Position Vectors: Displacement of a Point from the Origin A displacement vector whose tail is at the origin is called a position vector. Thus, any point (x0 , y0 , z0 ) in space has associated with it the position vector ~r 0 = x0~i + y0~j + z0~k . (See Figure 13.11.) In general, a position vector gives the displacement of a point from the origin. z z (x0 , y0 , z0 ) \u001b P1 P2 P2 = (x2 , y2 , z2 ) z0 ~r 0 P1 = (x1 , y1 , z1 ) y y0 y \u001b 690 x0 x Figure 13.10: The displacement vector P1 P2 = (x2 x1 )~i +(y2 y1 )~j +(z2 z1 )~k x Figure 13.11: The position vector ~r 0 = x0~i + y0~j + z0~k The Components of the Zero Vector The zero displacement vector has magnitude equal to zero and is written ~0 . So ~0 = 0~i + 0~j + 0~k . The Magnitude of a Vector in Components For a vector, ~v = v1~i + v2~j , the Pythagorean theorem is used to find its magnitude, k~v k. (See Figure 13.12.) y k~v k = Length = p v12 + v22 ~v v2 v1 x Figure 13.12: Magnitude, k~v k, of a 2-dimensional vector, ~v In 3-dimensions, for a vector ~v = v1~i + v2~j + v3~k , we have Magnitude of ~v = k~v k = Length of the arrow = For instance, if ~v = 3~i 4~j + 5~k , then k~v k = p q v12 + v22 + v32 . 32 + (4)2 + 52 = 50. 13.1 DISPLACEMENT VECTORS 691 Addition and Scalar Multiplication of Vectors in Components Suppose the vectors ~v and w ~ are given in components: ~v = v1~i + v2~j + v3~k and w ~ = w1~i + w2~j + w3~k . Then ~v + w ~ = (v1 + w1 )~i + (v2 + w2 )~j + (v3 + w3 )~k , and ~v = v1~i + v2~j + v3~k . Figures 13.13 and 13.14 illustrate these properties in two dimensions. Finally, ~v w ~ = ~v + (1)w ~ , so we can write ~v w ~ = (v1 w1 )~i + (v2 w2 )~j + (v3 w3 )~k . v1 ~v v2 2~v ~v + w ~ w ~ w ~ w2 v1 2v2 w1 ~v v2 v1 Figure 13.13: Sum ~v + w ~ in components v2 ~v 3~v 3v2 2v1 3v1 Figure 13.14: Scalar multiples of vectors showing ~v , 2~v , and 3~v How to Resolve a Vector into Components You may wonder how we find the components of a 2-dimensional vector, given its length and direction. Suppose the vector ~v has length v and makes an angle of with the x-axis, measured counterclockwise, as in Figure 13.15. If ~v = v1~i + v2~j , Figure 13.15 shows that v1 = v cos and v2 = v sin . Thus, we resolve ~v into components by writing ~v = (v cos )~i + (v sin )~j . Vectors in 3-space are resolved using direction cosines; see Problem 57 on page 719. y \u001b v v sin \u001b x v cos Figure 13.15: Resolving a vector: ~v = (v cos )~i + (v sin )~j Example 4 Solution Resolve ~v into components if k~v k = 2 and = /6. We have ~v = 2 cos(/6)~i + 2 sin(/6)~j = 2( 3/2)~i + 2(1/2)~j = 3~i + ~j . 692 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS Unit Vectors A unit vector is a vector whose magnitude is 1. The vectors ~i , ~j , and ~k are unit vectors in the directions of the coordinate axes. It is often helpful to find a unit vector in the same direction as a given vector ~v . Suppose that k~v k = 10; a unit vector in the same direction as ~v is ~v /10. In general, a unit vector in the direction of any nonzero vector ~v is ~u = ~v . k~v k Example 5 Find a unit vector, ~u , in the direction of the vector ~v = ~i + 3~j . Solution If ~v = ~i + 3~j , then k~v k = 1 2 + 32 = 10. Thus, a unit vector in the same direction is given by 1 1 ~v 3 ~u = = (~i + 3~j ) = ~i + ~j 0.32~i + 0.95~j . 10 10 10 10 Example 6 Find a unit vector at the point (x, y, z) that points radially outward away from the origin. Solution The vector from the origin to (x, y, z) is the position vector ~r = x~i + y~j + z~k . Thus, if we put its tail at (x, y, z) it will point away from the origin. Its magnitude is p k~r k = x2 + y 2 + z 2 , so a unit vector pointing in the same direction is x y z ~r x~i + y~j + z~k ~k . ~i + p ~j + p =p =p 2 2 2 2 2 2 2 2 2 2 k~r k x +y +z x +y +z x +y +z x + y2 + z 2 Exercises and Problems for Section 13.1 Exercises Resolve the vectors in Exercises 1-6 into components. 1. 2. y 3 y 3 d~ 2 2 ~a 1 ~b w ~ 1 ~v 1 1 1 2 2 ~c ~a x x 2 ~b 1 3 2 ~e 1 3 4 13.1 DISPLACEMENT VECTORS 3. A vector starting at the point Q = (4, 6) and ending at the point P = (1, 2). Find the length of the vectors in Exercises 15-19. 4. A vector starting at the point P = (1, 2) and ending at the point Q = (4, 6). 15. ~v = ~i ~j + 2~k ~v 3 y 2 x \u001b \u001b y 1 ~b = 3~i + 5~j + 4~k , ~c = ~i + 6~j , \u001b \u001b 1 17. ~v = ~i ~j + 3~k 18. ~v = 7.2~i 1.5~j + 2.1~k 19. ~v = 1.2~i 3.6~j + 4.1~k ~a = 2~j + ~k , \u001b ~c 16. ~z = ~i 3~j ~k For Exercises 20-25, perform the indicated operations on the following vectors: ~ u 2 \u001b \u001b \u001b x \u001b f~ d~ \u001b 1 \u001b ~e ~a ~b z 6. \u001b z 5. 693 ~ x = 2~i + 9~j , ~ y = 4~i 7~j , ~ z = ~i 3~j ~k . For Exercises 7-14, perform the indicated computation. 7. (4~i + 2~j ) (3~i ~j ) 8. (~i + 2~j ) + (3)(2~i + ~j ) 9. 4(~i 2~j ) 0.5(~i ~k ) 10. 2(0.45~i 0.9~j 0.01~k ) 0.5(1.2~i 0.1~k ) 11. (3~i 4~j + 2~k ) (6~i + 8~j ~k ) 12. (4~i 3~j + 7~k ) 2(5~i + ~j 2~k ) 13. (0.6~i + 0.2~j ~k ) + (0.3~i + 0.3~k ) 14. 1 ~ (2i 2 20. 4~z 21. 5~a + 2~b 23. 2~c + ~ x 24. 2~a + 7~b 5~z 25. k~ y ~x k 22. ~a + ~z 26. (a) Draw the position vector for ~v = 5~i 7~j . (b) What is k~v k? (c) Find the angle between ~v and the positive x-axis. 27. Find the unit vector in the direction of 0.06~i 0.08~k . ~j + 3~k ) + 3(~i 16 ~j + 21 ~k ) Problems 28. Find a unit vector in the opposite direction to ~v = 2~i ~j 11~k . 29. Find the value(s) of a making ~v = 5a~i 3~j parallel to w ~ = a2~i + 6~j . 30. Find a vector with length 2 that points in the same direction as ~i ~j + 2~k . 31. (a) Find a unit vector from the point P = (1, 2) and toward the point Q = (4, 6). (b) Find a vector of length 10 pointing in the same direction. 34. (a) From Figure 13.16, read off the coordinates of the five points, A, B, C, D, E, and thus resolve into components the following two vectors: ~ u = (2.5)AB + (0.8)CD, ~v = (2.5)BA (0.8)CD (b) What is the relation between ~ u and ~v ? Why was this to be expected? y 3 32. If north is the direction of the positive y-axis and east is the direction of the positive x-axis, give the unit vector pointing northwest. 2 33. Resolve the following vectors into components: A (a) The vector in 2-space of length 2 pointing up and to the right at an angle of /4 with the x-axis. (b) The vector in 3-space of length 1 lying in the xzplane pointing upward at an angle of /6 with the positive x-axis. D 4 B E 1 C 1 2 3 4 5 6 x 7 Figure 13.16 35. Find the components of a vector p ~ which has the same direction as EA in Figure 13.16 and whose length equals two units. 694 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 36. For each of the four statements below, answer the following questions: Does the statement make sense? If yes, is it true for all possible choices of ~a and ~b ? If no, why not? (a) ~a + ~b = ~b + ~a (b) ~a + k~b k = k~a + ~b k ~ ~ (c) kb + ~a k = k~a + b k (d) k~a + ~b k = ~ k~a k + kb k. 37. Two adjacent sides of a regular hexagon are given as the vectors ~ u and ~v in Figure 13.17. Label the remaining sides in terms of ~ u and ~v . 42. Figure 13.18 shows a molecule with four atoms at O, A, B and C. Verify that every atom in the molecule is 2 units away from every other atom. z p C(1, 1/ 3, 2 2/3) 0 A(2, 0, 0) x y B(1, 3, 0) Figure 13.18 ~v ~ u Figure 13.17 38. For what values of t are the following pairs of vectors parallel? 6~i + 8~j + 3~k (a) 2~i + (t2 + 23 t + 1)~j + t~k , (b) t~i + ~j + (t 1)~k , 2~i 4~j + ~k (c) 2t~i + t~j + t~k , 6~i + 3~j + 3~k . 39. Find all vectors ~v in 2 dimensions having k~v k = 5 and the ~i -component of ~v is 3~i . 40. Find all vectors ~v in the plane such that k~v k = 1 and k~v + ~i k = 1. 41. A truck is traveling due north at 30 km/hr approaching a crossroad. On a perpendicular road a police car is traveling west toward the intersection at 40 km/hr. Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the police car. 13.2 43. Show that the medians of a triangle intersect at a point of the way along each median from the side it bisects. 1 3 44. In the game of laser tag, you shoot a harmless laser gun and try to hit a target worn at the waist by other players. Suppose you are standing at the origin of a three dimensional coordinate system and that the xy-plane is the floor. Suppose that waist-high is 3 feet above floor level and that eye level is 5 feet above the floor. Three of your friends are your opponents. One is standing so that his target is 30 feet along the x-axis, the other lying down so that his target is at the point x = 20, y = 15, and the third lying in ambush so that his target is at a point 8 feet above the point x = 12, y = 30. (a) If you aim with your gun at eye level, find the vector from your gun to each of the three targets. (b) If you shoot from waist height, with your gun one foot to the right of the center of your body as you face along the x-axis, find the vector from your gun to each of the three targets. VECTORS IN GENERAL Besides displacement, there are many quantities that have both magnitude and direction and are added and multiplied by scalars in the same way as displacements. Any such quantity is called a vector and is represented by an arrow in the same manner we represent displacements. The length of the arrow is the magnitude of the vector, and the direction of the arrow is the direction of the vector. Velocity Versus Speed The speed of a moving body tells us how fast it is moving, say 80 km/hr. The speed is just a number; it is therefore a scalar. The velocity, on the other hand, tells us both how fast the body is moving and the direction of motion; it is a vector. For instance, if a car is heading northeast at 80 km/hr, then its velocity is a vector of length 80 pointing northeast. The velocity vector of a moving object is a vector whose magnitude is the speed of the object and whose direction is the direction of its motion