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Unformatted text preview: 3 Vectors CHAPTER OUTLINE 3.1 Coordinate Systems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors 3.4 Components of a Vector and Unit Vectors ANSWERS TO QUESTIONS Q3.1 Only force and velocity are vectors. None of the other quantities requires a direction to be described. The answers are (a) yes (b) no (c) no (d) no (e) no (f ) yes (g) no. Q3.2 The books displacement is zero, as it ends up at the point from which it started. The distance traveled is 6.0 meters. *Q3.3 The vector 2 r D 1 will be twice as long as r D 1 and in the opposite direction, namely northeast. Adding r D 2 , which is about equally long and southwest, we get a sum that is still longer and due east, choice (a). *Q3.4 The magnitudes of the vectors being added are constant, and we are considering the magnitude onlynot the directionof the resultant. So we need look only at the angle between the vectors being added in each case. The smaller this angle, the larger the resultant magnitude. Thus the ranking is c = e > a > d > b. *Q3.5 (a) leftward: negative. (b) upward: positive (c) rightward: positive (d) downward: negative (e) Depending on the signs and angles of r A and r B , the sum could be in any quadrant. (f) Now r A will be in the fourth quadrant, so + r r A B will be in the fourth quadrant. *Q3.6 (i) The magnitude is 10 10 2 2 + m s / , answer (f ). (ii) Having no y component means answer (a). *Q3.7 The vertical component is opposite the 30 angle, so sin 30 = (vertical component)/50 m and the answer is (h). *Q3.8 Take the difference of the coordinates of the ends of the vector. Final fi rst means head end fi rst. (i) 4 2 = 6 cm, answer ( j) (ii) 1 ( 2) = 3 cm, answer (c) Q3.9 (i) If the direction-angle of r A is between 180 degrees and 270 degrees, its components are both negative: answer (c). If a vector is in the second quadrant or the fourth quadrant, its components have opposite signs: answer (b) or (d). Q3.10 Vectors r A and r B are perpendicular to each other. Q3.11 No, the magnitude of a vector is always positive. A minus sign in a vector only indicates direction, not magnitude. Q3.12 Addition of a vector to a scalar is not defi ned. Think of numbers of apples and of clouds. 45 13794_03_ch03_p045-064.indd 45 13794_03_ch03_p045-064.indd 45 11/28/06 4:40:06 PM 11/28/06 4:40:06 PM SOLUTIONS TO PROBLEMS Section 3.1 Coordinate Systems P3.1 x r = = ( ) = ( ) ( ) = cos cos . 5 50 240 5 50 0 5 2 7 . m . m . 5 m y r = = ( ) = ( ) ( ) = sin sin 5 50 240 5 50 0 8 4 . m . m . 66 .76 m P3.2 (a) x r = cos and y r = sin , therefore x 1 2 50 30 0 = ( ) . m . cos , y 1 2 50 30 0 = ( ) . m . sin , and x y 1 1 2 17 1 25 , . , . m ( ) = ( ) x 2 3 80 120 = ( ) . c o s m , y 2 3 80 120 = ( ) . s i n m , and x y 2 2 1 90 3 29 , . , . m ( ) = ( ) (b) d x y = + = + = ( ) ( ) . . . 2 2 2 2 4 07 2 04 4 55 m m P3.3 The x distance out to the y is 2.00 m and the y distance up to the...
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