Questions: Vector Operations: Addition, Subtraction, and Scalar Multiplication
5 questions to test your understanding
Score: 0 / 5
Question 1 Multiple Choice
Vectors u = (3, 4) and v = (1, -2) are given. What is u + v?
A(4, 2)
B(2, 6)
C(3, -8)
D(4, 6)
Vector addition is component-wise: (3+1, 4+(-2)) = (4, 2). The common error is adding magnitudes (|u| = 5, |v| = √5) and getting a scalar — but that ignores direction entirely. Each component is handled independently.
Question 2 Multiple Choice
You want to rotate vector (1, 0) so it points in the direction (0, 1). Which scalar multiple of (1, 0) achieves this?
AMultiply by -1 to reverse direction, then the perpendicular component appears
BMultiply by a sufficiently large positive scalar
CNo scalar multiple of (1, 0) can ever point in the direction (0, 1)
DMultiply by i (the imaginary unit) to rotate 90°
Scalar multiplication c·(a, b) = (ca, cb) scales both components by the same factor c. It can stretch, shrink, or reverse a vector, but it cannot rotate it to any arbitrary direction — a negative scalar reverses direction (180°), and a positive scalar preserves it. To change direction by any angle other than 0° or 180°, you need vector addition. This is one of the most important constraints of scalar multiplication.
Question 3 True / False
Multiplying a vector by the scalar -3 reverses its direction and triples its magnitude.
TTrue
FFalse
Answer: True
For c·(a, b) = (-3a, -3b): the magnitude scales by |c| = |-3| = 3 (tripled), and the negative sign reverses direction (equivalent to a 180° rotation). Both effects happen simultaneously. This is precisely what scalar multiplication can do — scale and optionally reverse — but nothing else with respect to direction.
Question 4 True / False
Vector subtraction is commutative: u - v equals v - u.
TTrue
FFalse
Answer: False
u - v = u + (-v) points from the tip of v to the tip of u. Swapping gives v - u = v + (-u), which points in exactly the opposite direction. Geometrically, u - v and v - u are negatives of each other. This is analogous to scalar subtraction: 5 - 3 ≠ 3 - 5. Vector subtraction is not commutative.
Question 5 Short Answer
Explain geometrically what the vector u - v represents, and how its direction differs from v - u.
Think about your answer, then reveal below.
Model answer: u - v is the displacement vector pointing from the tip of v to the tip of u (when both are drawn from the same origin). It represents 'how you get from v to u.' v - u points in the exact opposite direction — from the tip of u to the tip of v. They have the same magnitude but opposite directions, so u - v = -(v - u).
The geometric interpretation of u - v comes from rewriting it as: 'start at v, what do you add to reach u?' That answer is u - v. This is why u - v is sometimes written as the 'vector from v to u.' Confusing the order is a persistent error; always ask 'which tip am I pointing toward?'