Biologia

3
Vectors
CHAPTER OUTLINE
3.1 3.2 3.3 3.4 Coordinate Systems Vector and Scalar Quantities Some Properties of Vectors 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. The book’s displacement is zero, asit ends up at the point from which it started. The distance traveled is 6.0 meters. The vector −2D1 will be twice as long as D1 and in the opposite direction, namely northeast. Adding D 2 , which is about equally long and southwest, we get a sum that is still longer and due east, choice (a). The magnitudes of the vectors being added are constant, and we are considering the magnitude only—not thedirection—of 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.

Q3.2

*Q3.3

*Q3.4

*Q3.5

(a) leftward: negative. (b) upward: positive (c) rightward: positive (d) downward: negative (e) Depending on the signs and angles of A and B, the sum couldbe in any quadrant. (f) Now − A will be in the fourth quadrant, so − A + B will be in the fourth quadrant. (i) The magnitude is 10 2 + 10 2 m s, answer (f ). (ii) Having no y component means answer (a). The vertical component is opposite the 30° angle, so sin 30° = (vertical component)/50 m and the answer is (h). Take the difference of the coordinates of the ends of the vector. Final first meanshead end first. (i) −4 − 2 = −6 cm, answer ( j) (ii) 1 − (−2) = 3 cm, answer (c) (i) If the direction-angle of 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). Vectors A and B are perpendicular to each other. No, the magnitude of a vector isalways positive. A minus sign in a vector only indicates direction, not magnitude. Addition of a vector to a scalar is not defined. Think of numbers of apples and of clouds.

*Q3.6 *Q3.7

*Q3.8

Q3.9

Q3.10 Q3.11

Q3.12

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Chapter 3

SOLUTIONS TO PROBLEMS
Section 3.1 P3.1 Coordinate Systems

x = r cos θ = ( 5.50 m ) cos240° = ( 5.50 m ) ( −0.5) = −2.75 m y = r sin θ = ( 5.50 m ) sin 240° = ( 5.50 m ) ( −0.866 ) = −4.76 m

P3.2

(a)

x = r cos θ and y = r sin θ , therefore x1 = ( 2.50 m ) cos 30.0°, y1 = ( 2.50 m ) sin 30.0°, and

( x1 , y1 ) = ( 2.17, 1.25) m
x2 = ( 3.80 m ) cos 120°, y2 = ( 3.80 m ) sin 120°, and

( x2 , y2 ) = ( −1.90,
(b) P3.3

3.29 ) m

d = ( ∆ x )2 + ( ∆ y )2 = 4.07 2 +2.04 2 m = 4.55 m

The x distance out to the fly is 2.00 m and the y distance up to the fly is 1.00 m. (a) We can use the Pythagorean theorem to find the distance from the origin to the fly. distance = x 2 + y 2 = ( 2.00 m ) + (1.00 m ) = 5.00 m 2 = 2.24 m
2 2

(b) P3.4

⎛ 1⎞ θ = tan −1 ⎜ ⎟ = 26.6°; r = 2.24 m, 26.6° ⎝ 2⎠ 2.00 = 2.31 cos 30.0°

We have 2.00 = r cos 30.0° r=

and y = r sin30.0° = 2.31 sin 30.0° = 1.15 . P3.5 ⎛ y⎞ We have r = x 2 + y 2 and θ = tan −1 ⎜ ⎟ . ⎝ x⎠ (a) The radius for this new point is

( − x )2 + y 2
and its angle is

= x 2 + y2 = r

y tan −1 ⎛ ⎞ = 180° − θ . ⎝ −x⎠ (b) (−2 x )2 + (−2 y )2 = 2 r This point is in the third quadrant if ( x, y ) is in the first quadrant

or in the fourth quadrant if ( x, y ) is in the second quadrant. It is at an angleof 180° + θ . (c) or in the third quadrant if ( x, y ) is in the second quadrant. It is at an angle of − θ .

( 3x )2 + (−3y )2 = 3r This point is in the fourth quadrant if ( x, y ) is in the first quadrant

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Vectors

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Section 3.2 Section 3.3 P3.6

Vector and Scalar Quantities Some Properties of Vectors

− R = 310 km at 57° S...