6.4 Vectors and Dot Products

Transcription

1 6.4 Vectors and Dot Products Copyright Cengage Learning. All rights reserved.

2 What You Should Learn Find the dot product of two vectors and use the properties of the dot product. Find the angle between two vectors and determine whether two vectors are orthogonal. Write vectors as the sums of two vector components. Use vectors to find the work done by a force. 2

3 The Dot Product of Two Vectors 3

4 The Dot Product of Two Vectors So far you have studied two vector operations vector addition and multiplication by a scalar each of which yields another vector. In this section, you will study a third vector operation, the dot product. This product yields a scalar, rather than a vector. 4

5 The Dot Product of Two Vectors 5

6 Example 2 Using Properties of Dot Products Let u = 1, 3, v = 2, 4, and w = 1, 2. Use the vectors and the properties of the dot product to find (a) (u v)w, (b) u 2v, and (c) u. Solution: Begin by finding the dot product of u and v and the dot product of u and u. u v = 1, 3 2, 4 = 1(2) + 3( 4) = 14 6

7 Example 2 Solution cont d u u = 1, 3 1, 3 = 1( 1) + 3(3) = 10 a. (u v)w = 14 1, 2 = 14, 28 7

8 Example 2 Solution cont d b. u 2v = 2(u v) = 28 c. Because u 2 = u u = 10, it follows that 8

9 The Angle Between Two Vectors 9

10 The Angle Between Two Vectors The angle between two nonzero vectors is the angle, 0, between their respective standard position vectors, as shown in the figure. This angle can be found using the dot product. (Note that the angle between the zero vector and another vector is not defined.) 10

11 The Angle Between Two Vectors 11

12 Example 3 Finding the Angle Between Two Vectors Find the angle between u = 4, 3 and v = 3, 5. Solution: 12

13 Example 3 Solution cont d This implies that the angle between the two vectors is as shown below. 13

14 The Angle Between Two Vectors The terms orthogonal and perpendicular mean essentially the same thing meeting at right angles. Even though the angle between the zero vector and another vector is not defined, it is convenient to extend the definition of orthogonality to include the zero vector. In other words, the zero vector is orthogonal to every vector u because 0 u = 0. 14

15 Example 4 Determining Orthogonal Vectors Are the vectors orthogonal? u = 2, 3 and v = 6, 4 Solution: Find the dot product of the two vectors. u v = 2, 3 6, 4 = 2(6) + ( 3)(4) = 0 15

16 Example 4 Solution cont d Because the dot product is 0, the two vectors are orthogonal, as shown below. 16

17 Determining Parallel Vectors Two vectors, v and w, are parallel if there is a scalar, c, so that cv = w 17

18 Homework: Page 440 # s 13, 15, 17, 23, 25, 27, 35, odd 18

19 Finding Vector Components 19

20 Finding Vector Components You have already seen applications in which two vectors are added to produce a resultant vector. Many applications in physics and engineering pose the reverse problem decomposing a given vector into the sum of two vector components. Consider a boat on an inclined ramp, as shown in Figure The force F due to gravity pulls the boat down the ramp and against the ramp. Figure

21 Finding Vector Components These two orthogonal forces, w 1 and w 2, are vector components of F. That is, F = w 1 + w 2. Vector components of F The negative of component w 1 represents the force needed to keep the boat from rolling down the ramp, and w 2 represents the force that the tires must withstand against the ramp. 21

22 Finding Vector Components A procedure for finding w 1 and w 2 is shown below. 22

23 Finding Vector Components is acute. is obtuse. Figure

24 Finding Vector Components From the definition of vector components, notice that you can find the component w 2 once you have found the projection of u onto v. To find the projection, you can use the dot product, as follows. u = w 1 + w 2 u = cv + w 2 u v = (cv + w 2 ) v w 1 is a scalar multiple of v. Take dot product of each side with v. u v = cv v + w 2 v u v = c v w 2 and v are orthogonal. 24

25 Finding Vector Components So, and 25

26 Example 5 Decomposing a Vector into Components Find the projection of u = 3, 5 onto v = 6, 2. Then write u as the sum of two orthogonal vectors, one of which is. 26

27 Example 5 Solution The projection of u onto v is as shown in Figure Figure

28 Example 5 Solution cont d The other component, w 2, is So, u = w 1 + w 2 = 3, 5. 28

29 Work 29

30 Work The work W done by a constant force F acting along the line of motion of an object is given by W = (magnitude of force)(distance) = F as shown in Figure Force acts along the line of motion. Figure

31 Work If the constant force F is not directed along the line of motion (see Figure 6.30), Force acts at angle with the line of motion. Figure 6.30 then the work W done by the force is given by Projection form for work Dot product form for work 31

32 Work 32

33 Example 7 Finding the Work Done To close a barn s sliding door, a person pulls on a rope with a constant force of 50 pounds at a constant angle of 60, as shown in Figure Figure

34 Example 7 Finding the Work Done cont d Find the work done in moving the door 12 feet to its closed position. Solution: Using a projection, you can calculate the work as follows. = 300 foot-pounds 34

35 Example 7 Solution cont d So, the work done is 300 foot-pounds. You can verify this result by finding the vectors F and and calculating their dot product. 35