HW4 even solutions

# HW4 even solutions - Homework 4 Solutions 13.1.2 i 2j 6i 3j...

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Homework 4 Solutions 13.1.2 ~ i + 2 ~ j - 6 ~ i - 3 ~ j = - 5 ~ i - ~ j . 13.1.6 k ~v k = p 1 2 + ( - 1) 2 + 2 2 = 6 . 13.1.12 k ~ z k = p (1) 2 + ( - 3) 2 + ( - 1) 2 = 1 + 9 + 1 = 11 . 13.1.14 2 ~a + 7 ~ b - 5 ~ z = 2(2 ~ j + ~ k ) + 7( - 3 ~ i + 5 ~ j + 4 ~ k ) - 5( ~ i - 3 ~ j - ~ k ) = (4 ~ j + 2 ~ k ) + ( - 21 ~ i + 35 ~ j + 28 ~ k ) - (5 ~ i - 15 ~ j - 5 ~ k ) = ( - 21 - 5) ~ i + (4 + 35 + 15) ~ j + (2 + 28 + 5) ~ k = - 26 ~ i + 54 ~ j + 35 ~ k 13.1.20 ~u = ~ i + ~ j + 2 ~ k and ~v = - ~ i + 2 ~ k . 13.2.6 We need to calculate the length of each vector: k 21 ~ i + 35 ~ j k = p 21 2 + 35 2 = 1666 40 . 8 , k 40 ~ i k = 40 2 = 40 . So the first car is faster. 13.2.8 In components, we have ~v = 10 cos(45 ) ~ i - 10 sin(45 ) ~ j = (5 2) ~ i - (5 2) ~ j = 7 . 07 ~ i - 7 . 07 ~ j . Notice that the coefficient in the ~ j -direction must be negative. The components are 5 2 ~ i and - 5 2 ~ j . 13.2.12 (a) The velocity vector for the boat ~ b = 25 ~ i and the velocity vector for the current is ~ c = - 10 cos(45 ) ~ i - 10 sin(45 ) ~ j = - 7 . 07 ~ i - 7 . 07 ~ j. The actual velocity of the boat is ~ b + ~ c = 17 . 93 ~ i - 7 . 07 ~ j. (b) k ~ b + ~ c k = 19 . 27 km/hr. (c) We see in the figure below that tan θ = 7 . 07 17 . 93 , so θ = 21 . 52 south of east. 7.07 17.93 θ b c + c b 1

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13.2.14 The velocity vector of the plane with respect to the air has the form ~v = a ~ i + 80 ~ k where k ~v k = 480 . (See figure below.) Therefore a 2 + 80 2 = 480 so a = 480 2 - 80 2 473 . 3 km/hr. We conclude that ~v 473 . 3 ~ i + 80 ~ k .
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