Solutions to Homework 3

12.1

#1 [5] p⃗ = 2w⃗, q⃗ = −u⃗, r⃗ = u⃗ + w⃗, s⃗ = p⃗ + q⃗ = 2w⃗ − u⃗, t⃗ = u⃗ − w⃗.

#7 Q⃗P = −3i⃗ + 4j⃗.

#40 a) \sqrt{2}i⃗ + \sqrt{2}j⃗,

b) \sqrt{3/4}i⃗ + 1/2k⃗.

#41 < 2/\sqrt{6}, −2/\sqrt{6}, 4/\sqrt{6} >.

12.2

#5 [3] At P a small vector pointing up, at Q a big vector pointing left, and at R a medium size vector pointing down.

#16 The plane should head roughly 3.4^{o} to the north.

#17 .8v cos 60i⃗ + .8v sin 60j⃗.

#21 Take the triangle with sides u⃗, v⃗, u⃗ + v⃗ and scale it by a factor of a. The result is a triangle with sides au⃗, av⃗, a(u⃗ + v⃗) and so au⃗ + av⃗ = a(u⃗ + v⃗).

#* [4]

a) The object is at (π, 2), its velocity is (2πi⃗) + (2πi⃗) = 4πi⃗ and so its speed is 4π m/min.

b) The object is at (π − 1, 1), its velocity is (2πi⃗) + (2πj⃗) = 2π(i⃗+ j⃗) and so its speed is 2π\sqrt{2} m/min.

12.3

#16 2x + 4y − 3z = 5.

#18 [4] a) n⃗ =< 3, −1, −1 >

b) Anything perpendicular to n⃗ works, for example < 1, 3, 0 >.

#27 c^{2} = ∥u⃗−v⃗∥^{2} = (u⃗−v⃗)·(u⃗−v⃗) = u⃗·u⃗−2u⃗·v⃗+v⃗·v⃗ = ∥u⃗∥^{2}+∥v⃗∥^{2}−2∥u⃗∥∥v⃗∥ cos C = a^{2} + b^{2} − 2ab cos C.

#35 If u⃗ = 0 both sides are 0, so we may assume that u⃗ is not 0. We can also assume that v⃗_{∥} +w⃗_{∥} = (v⃗ +w⃗)_{∥}, which means that \frac{v⃗·u⃗}{u⃗·u⃗} u⃗+ \frac{w⃗·u⃗}{u⃗·u⃗} u⃗ = \frac{(v⃗+w⃗)·u⃗}{u⃗·u⃗} u⃗ so multiplying both sides by u⃗·u⃗ and eliminating the vector u⃗, we get that v⃗·u⃗+w⃗·u⃗ = (v⃗+w⃗)·u⃗.

#36 [4] The angle between the English and the Bantu is .1723 and the angle between the English and the Koreans is .2553 so that the English are genetically closer to the Bantus than to the Koreans.