1
1.3
14: Find the angle θ between the vectors (a) in radians and (b) in degrees
~u = 3~i + 2~j + ~k,
Solution: cos θ =
~
u·~
v
k~
ukk~
vk
~v = 2~i − 3~j.
= 0. So θ =
π
2
= 90◦ .
42: (a) Find the projection of ~u onto ~v , and (b) Find the vector component of
~u orthogonal to ~v .
~u = ~i + 4~k, ~v = 3~i + 2~k.
Solution:
(a) :
Proj~v ~u =
~u · ~v
11 ~
~v =
(3i + 2~k).
k~v k2
13
(b): Vector component orthogonal to ~v :
~u − Proj~v ~u =
−20~i + 30~j
.
13
57: Find the angle between the diagonal of a cube and one of its adjacent edges.
Show your work.
Solutions: Vector in the direction of diagonal ~u = h1, 1,
√1i. Vector along
adjacent side ~v = h1, 0, 0i. So cos θ = √13 and θ = Arccos(1/ 3) ≈ 54.7◦ .
62: A toy wagon is pulled by exerting a force of 25 pounds on a handle that
makes a 20◦ angle with the horizontal. Find the work done in pulling the wagon
50 feet.
Solutions: Work=25 × cos(20◦ ) × 50 = 1174.62 ft-lbs.
2
1.4
1-6: Find the cross product of unit vectors and sketch your result in pictures:
1: ~j × ~i 2: ~i × ~j 3: ~j × ~k 4: ~k × ~j 5: ~i × ~k 6: ~k × ~i.
Solution: 1: −~k 2: ~k 3: ~i 4: −~i 5: −~j 6: ~j.
10: Find (a) ~u × ~v , (b) ~v × ~u and (c) ~v × ~v .
~u = h3, −2, −2i,
~v = h1, 5, 1i.
Solution: (a): h8, −5, 17i, (b): h−8, 5, −17i, (c): 0.
16: Find ~u × ~v and show that it is orthogonal to both ~u and ~v .
~u = ~i + 6~j,
~v = −2~i + ~j + ~k.
Solution: ~u × ~v = h5, −3, 13i.
1
28: Find the area of the triangle with given vertices.
A(2, −3, 4),
B(0, 1, 2),
C(−1, 2, 0).
(Hint: 21 k~u × ~v k is the area of the triangle having ~u and ~v as adjacent sides.
−−→
−−→
Solution: AB = h−2, 4, −2i, AC = h−3, 5, −4i.
~i ~j ~k −−→ −−→ AB × AC = −2 4 −2 = h−6, −2, 2i.
−3 5 −4 Area(4ABC ) =
1 −−→ −−→ √
AB × AC = 11.
2
38: Use the triple scalar product to find the volume of the parallelpiped having
adjacent edges ~u, ~v and w.
~
~u = h1, 3, 1i,
Solution:
~v = h0, 6, 6i,
1 3
~u × ~v · w
~ = 0 6
−4 0
The volume of the parallelepiped=72.
2
w
~ = h−4, 0, −4i.
1
6
−4
= −72.