1
1.1
8. Find the vectors whose initial and terminal points are given. Show that ~u
and ~v are equivalent.
~u : (−4, −1), (11, −4);
~v : (10, 13), (25, 10)
Solution: ~u = h15, −3i, ~v = h15, −3i. So ~u = ~v .
28. The vector ~v and its initial point are given. Find the terminal point.
~v = h4, −9i; Initial point:(5, 3).
Solution: Terminal point is (9, −6).
36. Find the unit vector in the direction of ~v and verify that it has length 1.
~v = h−5, 15i
Solution: Unit vector in the direction of ~v is
3
~v
h−5, 15i
1
.
=
= −√ , √
k~v k
kh−5, 15ik
10
10
50. Find the component form of ~v given its magnitude and the angle it makes
with the positive x-axis.
k~v k = 5, θ = 120◦ .
D
√ E
Solution: ~v = k~v khcos θ, sin θi = − 52 , 5 2 3 .
84. A plane flies at a constant groundspeed of 400 miles per hour due east and
encounters a 50-miles-per-hour wind from the northwest. Find the airspeed and
compass direction that will allow the plane to maintain its groundspeed and
eastward direction. (Hint: The velocity of the plane is the sum of the windvelocity and the air-velocity. The air-velocity is the scalar multiplication of the
airspeed with the compass direction)
Air velocity
:
- Ground velocity
@
@
@
RWind
@
√
√
Solution: Ground velocity=h400, 0i. Wind velocity=h25 2, −25 2i. So
air velocity is
√
√
~v = h400 − 25 2, 25 2i ≈ h364.6, 35.4i.
The airspeed is k~v k = 366.4 miles per hour. The compass direction is
~v
≈ h0.995, 0.097i.
k~v k
√
√
Or equivalently, the angle θ is about arctan(25 2/(400 − 25 2)) ≈ 5.5◦ north
of due east.
1
2
1.2
6. Find the coordinates of the point. The point is located seven units in front
of the yz-plane, two units to the left of the xz-plane, and one unit below the
xy-plane.
Solution: (7, 2, −1).
26. Find the distance between the points.
(2, 2, 3), (4, −5, 6)
p
√
Solution: distance= (4 − 2)2 + (−5 − 2)2 + (6 − 3)2 = 62.
44. Complete the square to write the equation of the sphere in standard form.
Find the center and radius.
4x2 + 4y 2 + 4z 2 − 24x − 4y + 8z − 23 = 0.
Solution: Standard form:
1
(x − 3)2 + (y − )2 + (z + 1)2 = 16.
2
The center is (3, 1/2, −1). Radius is 4.
58. Given ~u = h1, 2, 3i, ~v = h2, 2, −1i, and w
~ = h4, 0, −4i. Find
1
~
~z = 5~u − 3~v − w.
2
Solution: ~z = h−3, 4, 20i.
84. Find the vector ~v with the given magnitude and the same direction as ~u
Magnitude: k~v k = 3; Direction: ~u = h1, 1, 1i.
√ √ √
Solution: ~v = k~v k k~~uuk = h 3, 3, 3i.
3
1.3
4. Find (a) ~u · ~v , (b) ~u · ~u, (c) k~uk2 , (d) (~u · ~v )~v , and (e) ~u · (2~v ).
~u = h−4, 8i,
~v = h7, 5i.
Solution: (a) 12. (b) 80. (c) 80. (d) h84, 60i. (e) 24.
10. Find the θ between the vectors (a) in radians and (b) in degrees.
~u = h3, 1i,
Solution:
cos θ =
So θ =
π
4
~v = h2, −1i
√
~u · ~v
2
=
.
k~ukk~v k
2
= 45◦ .
2