You compute c·v where v = (2, −3) and c = −2. A classmate says the answer is (4, −6) because 'a negative scalar flips the sign, so we take the absolute value of c for the result.' What is the correct result?
A(4, −6) — the classmate is correct
B(−4, 6) — each component is multiplied by c = −2, including its own sign
C(−4, −6) — negative times negative gives positive, so the second component stays negative
D(4, 6) — magnitude scales by |c| and direction is unchanged
Scalar multiplication applies c to every component directly: (−2)·2 = −4 and (−2)·(−3) = 6, giving (−4, 6). The classmate's error is treating the scalar's negativity as a 'sign flip' rather than a multiplication. Note that option C makes the classic sign error: (−2)(−3) = +6, not −6.
Question 2 Multiple Choice
Which of the following best describes what scalar multiplication does geometrically to a nonzero vector v when c = −1/2?
ARotates v by 180° without changing its length
BProduces a vector half as long as v, pointing in the same direction
CProduces a vector half as long as v, pointing in the opposite direction
DProjects v onto the axis of smallest magnitude
The scalar c = −1/2 has magnitude 1/2 (shrinks the vector) and is negative (reverses direction). So the result is antiparallel to v and half its original length. Option A is wrong because c = −1 would flip without scaling; c = −1/2 both flips and shrinks. Option B has the direction wrong.
Question 3 True / False
For any nonzero vector v and any scalar c, the vector cv is always parallel or antiparallel to v.
TTrue
FFalse
Answer: True
Scalar multiplication scales every component by the same factor c, which can only stretch, shrink, or flip the vector — it cannot rotate it or change its direction except to reverse it. The result is always collinear with v. This is precisely what makes scalar multiplication different from a rotation or projection.
Question 4 True / False
Multiplying a vector v by c = −1 produces a vector perpendicular to v.
TTrue
FFalse
Answer: False
Multiplying by −1 gives −v, which is antiparallel (opposite direction) to v — not perpendicular. Perpendicularity would mean the dot product is zero, but v · (−v) = −|v|² ≠ 0 for any nonzero vector. The confusion likely comes from thinking 'negative' means 'opposite' in a rotational sense, but in vector algebra it means 'same line, reversed direction.'
Question 5 Short Answer
Why does multiplying any vector v by c = 0 always produce the zero vector, regardless of what v is? What does this tell you about the role of the zero vector in a vector space?
Think about your answer, then reveal below.
Model answer: Since scalar multiplication applies c to every component, 0·vᵢ = 0 for each component vᵢ, yielding (0, 0, …, 0) = 0 regardless of v. This confirms the zero vector is the unique additive identity and the unique result of multiplying by the zero scalar. It plays a special role: every vector space must contain it, and it is reachable from any vector via scalar multiplication.
This is not just a computational fact but a structural one. The zero vector is special precisely because it sits at the 'collapse point' of scalar multiplication — scaling to zero. Recognizing that c = 0 always collapses to the zero vector, while c ≠ 0 preserves the direction (possibly reversed), helps distinguish when scalar multiplication is invertible (c ≠ 0, you can recover v by multiplying by 1/c) versus destructive (c = 0).