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quiz1-solutions

2 answer by elimination the following answers are

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2. Answer by elimination. The following answers are wrong: (e) If ~ b = - ~a have ~ c = 0, even though neither ~a nor ~ b vanish. (d) From right triangles we see that this can only be true if ~a is perpendicular to ~ b . (c) This can only happen if ~a is aligned with ~ b . Another way to see this is consider the explanation to why (e) is wrong: we could have ~ b = - ~a with neither ~a nor ~ b vanishing. (b) See previous. Hence (a) must be correct. 3. The construction in 1. above can be done analytically. To this end, we can use the definition of magnitude: | ~ c | 2 = ~ c · ~ c = ( ~a + ~ b ) · ( ~a + ~ b ) = | veca | 2 + | ~ b | 2 + 2 ~a · ~ b . We see that for fixed magnitudes | ~a | and | ~ b | , the magnitude | ~ c | varies only through the relative orientation θ of ~a and ~ b : | ~ c | 2 = | veca | 2 + | ~ b | 2 + 2 | ~a || ~ b | cos θ . Since - 1 cos θ 1 we have ( | ~a | - | ~ b | ) 2 ≤ | ~ c | 2 ( | ~a | + | ~ b | ) 2 . It follows that (a) is correct while the rest are wrong. 2
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2. Find the angle between the vectors (1 , 0) and ( - 7 , 7) (a) 135 4 (b) 45 (c) 2 . 36 (d) 0 . 79 (e) - 45 ANS: Two ways to answer this: 1. Clearly (1,0) is horizontal and to the right while (-7,7) is pointing up and to the left, making an angle of 45 to the horizontal. Hence the angle between them is 135 . A figure may help: (1,0) (-7,7) 45 2. The hard way: use cos θ = ~a · ~ b/ | ~a || ~ b | : cos θ = (1 , 0) · ( - 7 , 7) | (1 , 0) || ( - 7 , 7) | = - 1 2 and it follows that θ = 135 . 3
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3. Alice drives at a speed of 30 mph for 30 min heading north on Interstate-5. She stops for a 10 min coffee break and then speeds off for 45 miles in 20 min heading east on Interstate-8, until the California Highway Patrol stops her for speeding. Alice’s average speed for this trip is (a) 60 mph 4 (b) 47 mph (c) 82.5 mph (d) 55 mph (e) 72 mph ANS: Average speed, s avg , is total distance traveled divided by time interval. The time interval is easy: 30 + 10 + 20 min = 60 min = one hour. The distance travelled is from Δ x = v Δ t applied to the various straight light segments: for the first (30 mph) × (30 min) = 15 miles, for the third we are given it goes 45 miles. So s avg = (15 + 0 + 45) /
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