Chapter 6. Additional Topics in Trigonometry. 6.6 Vectors. Copyright 2014, 2010, 2007 Pearson Education, Inc.

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1 Chapter 6 Additional Topics in Trigonometry 6.6 Vectors Copyright 2014, 2010, 2007 Pearson Education, Inc. 1

2 Obectives: Use magnitude and direction to show vectors are equal. Visualize scalar multiplication, vector addition, and vector subtraction as geometric vectors. Represent vectors in the rectangular coordinate system. Perform operations with vectors in terms of i and. Find the unit vector in the direction of v. Write a vector in terms of its magnitude and direction. Solve applied problems involving vectors. Copyright 2014, 2010, 2007 Pearson Education, Inc. 2

3 Vectors Quantities that involve both a magnitude and a direction are called vector quantities, or vectors for short. Quantities that involve magnitude, but no direction, are called scalar quantities, or scalars for short. Copyright 2014, 2010, 2007 Pearson Education, Inc. 3

4 Directed Line Segments and Geometric Vectors A line segment to which a direction has been assigned is called a directed line segment. We call P the initial point and Q the terminal point. We denote this directed line segment by PQ. The magnitude of the directed line segment PQ is its length. We denote this by PQ. Geometrically, a vector is a directed line segment. Copyright 2014, 2010, 2007 Pearson Education, Inc. 4

5 Representing Vectors in Print and on Paper Copyright 2014, 2010, 2007 Pearson Education, Inc. 5

6 Equal Vectors In general, vectors v and w are equal if they have the same magnitude and the same direction. We write this v = w. Copyright 2014, 2010, 2007 Pearson Education, Inc. 6

7 Example: Showing that Two Vectors are Equal Show that u = v. Equal vectors have the same magnitude and the same direction. Use the distance formula to show that u and v have the same magnitude. 2 u ( x x ) y y (6 2) [ 2 ( 5)] Copyright 2014, 2010, 2007 Pearson Education, Inc. 7

8 Example: Showing that Two Vectors are Equal Show that u = v. Equal vectors have the same magnitude and the same direction. Use the distance formula to show that u and v have the same magnitude. 2 v ( x x ) y y (6 2) Copyright 2014, 2010, 2007 Pearson Education, Inc. 8

9 Example: Showing that Two Vectors are Equal Show that u = v. One way to show that u and v have the same direction is to find the slopes of the lines on which they lie. slope of u slope of v y y m x x y y m x x ( 5) Copyright 2014, 2010, 2007 Pearson Education, Inc. 9

10 Example: Showing that Two Vectors are Equal Show that u = v. u 5 slope of u v slope of v 4 3 The vectors have the same magnitude and direction. Thus, u = v. Copyright 2014, 2010, 2007 Pearson Education, Inc. 10

11 Scalar Multiplication The multiplication of a real number k and a vector v is called scalar multiplication. We write this product kv. Multiplying a vector by any positive real number (except 1) changes the magnitude of the vector but not its direction. Multiplying a vector by any negative number reverses the direction of the vector. Copyright 2014, 2010, 2007 Pearson Education, Inc. 11

12 Scalar Multiplication (continued) Copyright 2014, 2010, 2007 Pearson Education, Inc. 12

13 The Sum of Two Vectors The sum of u and v, denoted u + v is called the resultant vector. A geometric method for adding two vectors is shown in the figure. Here is how we find this vector: Position u and v, so that the terminal point of u coincides with the initial point of v. The resultant vector, u + v, extends from the initial point of u to the terminal point of v. Copyright 2014, 2010, 2007 Pearson Education, Inc. 13

14 The Difference of Two Vectors The difference of two vectors, v u, is defined as v u = v + ( u), where u is the scalar multiplication of u and 1, 1u. The difference v u is shown geometrically in the figure. Copyright 2014, 2010, 2007 Pearson Education, Inc. 14

15 The i and Unit Vectors Copyright 2014, 2010, 2007 Pearson Education, Inc. 15

16 Representing Vectors in Rectangular Coordinates Copyright 2014, 2010, 2007 Pearson Education, Inc. 16

17 Example: Representing a Vector in Rectangular Coordinates and Finding Its Magnitude Sketch the vector v = 3i 3 and find its magnitude. v ai v 2 v 3i 3 v ( x x ) y y a 3, b ( 3) initial point (0, 0) v = 3i terminal point (3, 3) Copyright 2014, 2010, 2007 Pearson Education, Inc. 17

18 Representing Vectors in Rectangular Coordinates (continued) Copyright 2014, 2010, 2007 Pearson Education, Inc. 18

19 Example: Representing a Vector in Rectangular Coordinates Let v be the vector from initial point P 1 = ( 1, 3) to terminal point P 2 = (2, 7). Write v in terms of i and. v ( x x ) i ( y y ) [2 ( 1)] i (7 3) 3i 4 P1 ( 1,3) P2 (2,7) (3,4) v = 3i + 4 Copyright 2014, 2010, 2007 Pearson Education, Inc. 19

20 Adding and Subtracting Vectors in Terms of i and Copyright 2014, 2010, 2007 Pearson Education, Inc. 20

21 Example: Adding and Subtracting Vectors If v = 7i + 3 and w = 4i 5, find the following vectors: a. v + w v w ( a a ) i ( b b ) (7 4) i [3 ( 5)] 11i 2 b. v w v w ( a a ) i ( b b ) (7 4) i [3 ( 5)] 3i 8 Copyright 2014, 2010, 2007 Pearson Education, Inc. 21

22 Scalar Multiplication with a Vector in Terms of i and Copyright 2014, 2010, 2007 Pearson Education, Inc. 22

23 Example: Scalar Multiplication If v = 7i + 10, find each of the following vectors: a. 8v kv ( ka) i ( kb) 8 v (8 7) i (8 10) 56i 80 b. 5v kv ( ka) i ( kb) 5 v ( 5 7) i ( 5 10) 35i 50 Copyright 2014, 2010, 2007 Pearson Education, Inc. 23

24 The Zero Vector Copyright 2014, 2010, 2007 Pearson Education, Inc. 24

25 Properties of Vector Addition Copyright 2014, 2010, 2007 Pearson Education, Inc. 25

26 Properties of Scalar Multiplication Copyright 2014, 2010, 2007 Pearson Education, Inc. 26

27 Unit Vectors A unit vector is defined to be a vector whose magnitude is one. Copyright 2014, 2010, 2007 Pearson Education, Inc. 27

28 Finding the Unit Vector that Has the Same Direction as a Given Nonzero Vector v Copyright 2014, 2010, 2007 Pearson Education, Inc. 28

29 Example: Finding a Unit Vector Find the unit vector in the same direction as v = 4i 3. Then verify that the vector has magnitude 1. v a b 2 2 v 4i 3 v ( 3) i Copyright 2014, 2010, 2007 Pearson Education, Inc. 29

30 Writing a Vector in Terms of Its Magnitude and Direction Copyright 2014, 2010, 2007 Pearson Education, Inc. 30

31 Example: Writing a Vector Whose Magnitude and Direction are Given The et stream is blowing at 60 miles per hour in the direction N45 E. Express its velocity as a vector v in terms of i and. 45, v 60 v v cos i v sin v v 60cos 45 i 60sin i i 30 2 The et stream can be expressed in terms of i and as 30 2i 30 2 Copyright 2014, 2010, 2007 Pearson Education, Inc. 31

32 Example: Application Two forces, F 1 and F 2, of magnitude 30 and 60 pounds, respectively, act on an obect. The direction of F 1 is N10 E and the direction of F 2 is N60 E. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force. N10 E Resultant force, F F 1 30 pounds N60 E F 2 60 pounds Copyright 2014, 2010, 2007 Pearson Education, Inc. 32

33 Example: Application (continued) F F cos i F sin cos80 i 30sin i F F cos i F sin cos30 i 60sin i 30 Resultant force, F N10 E F 1 30 pounds N60 E F 2 60 pounds Copyright 2014, 2010, 2007 Pearson Education, Inc. 33

34 Example: Application (continued) F1 5.21i F i 30 F F1 F2 (5.21i ) (51.96i 30 ) ( ) i ( ) 57.17i F a b cos 2 2 a F cos Copyright 2014, 2010, 2007 Pearson Education, Inc. 34

35 Example: Application Two forces, F 1 and F 2, of magnitude 30 and 60 pounds, respectively, act on an obect. The direction of F 1 is N10 E and the direction of F 2 is N60 E. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth of a degree, of the resultant force. The two given forces are equivalent to a single force of approximately pounds with a direction angle of approximately N10 E Resultant force, F F 1 30 pounds N60 E F 2 60 pounds Copyright 2014, 2010, 2007 Pearson Education, Inc. 35