Vectors — Practice
Some quantities need a direction as well as a size, and they do not add like numbers.
What these questions cover
These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Perpendicular — Slopes multiply to -1 — negative reciprocals.
- Vector — A quantity with both magnitude and direction.
- Scalar — A quantity with size only, such as mass or temperature.
- Magnitude — The length of the vector, written |v|.
- Resultant — The single vector that two or more vectors add up to.
Problem 1
For u=⟨ 3,1⟩ and v=⟨ 2,5⟩, find the first component of u+v.
Worked answer
5
3+2 = 5
Problem 2
For the same vectors, find the second component of u+v.
Worked answer
6
1+5 = 6
Problem 3
Find the magnitude of ⟨ 3, 4 ⟩.
Worked answer
5
√9+16 = 5
Problem 4
Find the magnitude of ⟨ -8, 6 ⟩.
Worked answer
10
√64+36 = 10
Problem 5
Find the magnitude of 3⟨ 1, 0 ⟩.
Worked answer
3
⟨ 3, 0 ⟩ has length 3
Problem 6
Find ⟨ 2, 3 ⟩ · ⟨ 4, 1 ⟩.
Worked answer
11
2(4)+3(1) = 8+3 = 11
Problem 7
Find ⟨ 5, -2 ⟩ · ⟨ 3, 4 ⟩.
Worked answer
7
15 + (-8) = 7
Problem 8
Are ⟨ 6, -2 ⟩ and ⟨ 1, 3 ⟩ perpendicular?
- Yes
- No
- Only if scaled
- Cannot tell
Worked answer
Yes
6(1)+(-2)(3) = 0
Zero means a right angle
Problem 9
A vector has magnitude 10 at 30°. Find its horizontal component.
Worked answer
5sqrt(3)
x
=
r
\
c
o
s
\
t
h
e
t
a
.
Problem 10
A vector has magnitude 10 at 30°. Find its vertical component.
Worked answer
5
10sin 30° = 5
Problem 11
For u=⟨ 7,2⟩ and v=⟨ 3,5⟩, find the second component of u-v.
Worked answer
-3
2-5 = -3
Problem 12
A plane flies north at 300 mph in a 40 mph wind from the west. Find the ground speed to one decimal place.
Worked answer
302.7
⟨ 40, 300 ⟩
√1600+90000 = √91600 ≈ 302.7
Problem 13
Two forces, 12 N east and 5 N north, act on a point. Find the magnitude of the resultant.
Worked answer
13
√144+25 = 13
Problem 14
Find the magnitude of the unit vector in the direction of ⟨ 6, 8 ⟩.
Worked answer
1
Dividing by the magnitude 10 gives ⟨ 0.6, 0.8 ⟩
Its length is 1 by construction
Problem 15
What does -2v do to v?
- Doubles its length and reverses it
- Doubles its length only
- Halves it
- Rotates it 90°
Worked answer
Doubles its length and reverses it
The 2 stretches, the minus reverses
Problem 16
Find k so that ⟨ 3, k ⟩ is perpendicular to ⟨ 2, 6 ⟩.
Worked answer
-1
6+6k = 0
k = -1