MATH 32A, BRIDGE 2013
M. Wang
Homework 3
Solution
1. Compute the dot product. (13.3 Exercises 1, 10)
a. h1, 2, 1i · h4, 3, 5i
Ans: h1, 2, 1i · h4, 3, 5i = 4 + 6 + 5 = 15
b. (3j + 2k) · (i − 4k)
Ans : (3j + 2k) · (i − 4k) = 3i · j − 12j · k + 2i · k − 8k · k = 0 − 0 + 0 − 8 = −8
2. Find the cosine of the angle between the vectors.(13.3 Exercises 19, 22)
a. h0, 3, 1i, h4, 0, 0i
Ans: cos θ =
h0,3,1i·h4,0,0i
kh0,3,1ikkh4,0,0ik
=0
b. 3i + k, i + j + k
Ans: cos θ =
(3i+k)(i+j+k)
k3i+kkki+j+kk
=
√ 4√
10 3
=
√
2 30
15
3. Simplify the expression. (13.3 Exercises 35, 37)
a. (v − w) · v + v · w
Ans: (v − w) · v + v · w = v2 − v · w + v · w = v2
b. (v + w) · v − (v + w) · w
Ans: (v + w) · v − (v + w) · w = v2 + w · v − v · w − w2 = v2 − w2
4. Show that if e and f are unit vectors such that ke + f k = 32 , then ke − f k =
e · f = 81 . (13.3 Exercises 47)
Proof. Since e and f are unit vectors, we know that
e · e = kek2 = f · f = kek2 = 1
ke + f k =
9
9
3
=⇒ (e + f )2 =
=⇒ e2 + f 2 + 2e · f =
2
4
4
1
7
2
2
=⇒ 2e · f =
=⇒ e + f − 2e · f =
4 √
4
7
=⇒ ke − f k =
2
5. Find the projection of u along v. (13.3 Exercises 51, 55)
1
√
7
2 .
Hint: Show that
a. u = h2, 5i, v = h1, 1i
7
7 7
Ans: uk = u·v
v
=
v·v
2 h1, 1i = h 2 , 2 i
b. u = 5i + 7j − 4k, v = k
−4
v
=
Ans: uk = u·v
v·v
1 k = −4k
6. Find the decomposition a = ak + a⊥ with respect to b. (13.3 Exercises 64, 65)
a. a = h2, −3i, b = h5, 0i
Ans: ak = 10
25 h5, 0i = h2, 0i, a⊥ = a − ak = h0, −3i
b. a = h4, −1, 0i, b = h0, 1, 1i
1
1
1 1
Ans: ak = −1
2 h0, 1, 1i = h0, − 2 , − 2 i, a⊥ = a − ak = h4, − 2 , 2 i
2