Vectors — Challenge
Some quantities need a direction as well as a size, and they do not add like numbers.
What these problems 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
What is (2,3)+(1,-1)?
- (3,2)
- (1,4)
- (3,4)
- (2,-3)
- (1,2)
Worked answer
(3,2)
Vectors are added one component at a time.
The first components give 2+1 = 3; the second give 3+(-1) = 2.
So the sum is (3,2).
Problem 2
What is the magnitude of the vector (3,4)?
- 3
- 4
- 5
- 7
- 25
Worked answer
5
The magnitude is the length of the arrow, found by Pythagoras on its components.
√32+42 = √25 = 5.
Adding the components gives 7, which is the distance travelled along the axes rather than the direct length.
Problem 3
For u=⟨ 2,3⟩ and v=⟨ 4,-1⟩, find |u+2v|.
- √101
- √85
- 11
- √13
- 10
Worked answer
√101
2v = ⟨ 8, -2 ⟩. Adding: u + 2v = ⟨ 10, 1 ⟩. Its magnitude is √100+1 = √101 ≈ 10.05.
Problem 4
What is the angle between ⟨ 1,0⟩ and ⟨ 1,1⟩?
- 45°
- 30°
- 60°
- 90°
- 135°
Worked answer
45°
The dot product is 1(1)+0(1) = 1. Also u·v = |u||v|cosθ = (1)(√2)cosθ. So cosθ = 1√2, giving θ = 45°.
Problem 5
A 100 N force acts at 60° above the horizontal. What is its horizontal component?
- 50 N
- 50√3 N
- 100 N
- 86.6 N
- 0 N
Worked answer
50 N
The horizontal component is |F|cosθ. 100cos 60° = 100(12). = 50 N. The vertical component is the larger one here, at 50√3 ≈ 86.6 N.
Problem 6
If |u| = 5, |v| = 12 and the two are perpendicular, what is |u+v|?
- 13
- 17
- 7
- 60
- √17
Worked answer
13
Perpendicular vectors form the legs of a right triangle, with the sum as the hypotenuse. |u+v| = √25+144 = √169. = 13. Adding the magnitudes would give 17, which is what happens only when they point the same way.
Problem 7
For which values of k are ⟨ k, 3⟩ and ⟨ 4, k⟩ parallel?
- ± 2√3
- ± 12
- ± 6
- 0 only
- ± 3
Worked answer
± 2√3
Parallel vectors are scalar multiples, so their components are in the same ratio: k4 = 3k. Cross-multiplying gives k2 = 12. So k = ± 2√3 ≈ ± 3.46.
Problem 8
A boat heads due north at 8 km/h across a river flowing east at 6 km/h. What is its speed relative to the bank?
- 10 km/h
- 14 km/h
- 2 km/h
- √14 km/h
- 48 km/h
Worked answer
10 km/h
The two velocities are perpendicular, so they combine as ⟨ 6, 8 ⟩. The magnitude is √36+64 = √100. = 10 km/h. The boat is faster than either figure alone, but slower than their sum.