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Learn Index · Vectors and Advanced Geometry Grade 11–12

Scalar multiplication of vectors

Multiplying a vector by a number changes its length (and flips it if negative).

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Stretching and shrinking vectors

Scalar multiplication multiplies a vector by a real number (a scalar) k.

k[x, y] = [kx, ky]

k Effect on the vector
k > 1 longer, same direction
0 < k < 1 shorter, same direction
k = 0 the zero vector
k < 0 opposite direction, length × |k|

Examples:

  • 3[2, −1] = [6, −3]
  • ½[8, 4] = [4, 2]
  • −2[1, 5] = [−2, −10]

Magnitude

The magnitude scales by the size of k: |kv| = |k| × |v|

If |v| = 5, then |3v| = 15 and |−2v| = 10.

Parallel vectors

Two vectors are parallel vectors if one is a scalar multiple of the other.

  • [2, 3] and [6, 9]: [6, 9] = 3[2, 3] ✓ parallel
  • [4, −2] and [−6, 3]: [−6, 3] = −1.5[4, −2] ✓ parallel (opposite direction)
  • [1, 2] and [2, 3]: no single k works ✗

Quick test in 2D: [a, b] ∥ [c, d] if ad − bc = 0.

Collinear points

Points A, B, C are collinear if AB⃗ and AC⃗ are parallel (they share point A).

Example: A(1, 1), B(3, 4), C(7, 10)

  • AB⃗ = [2, 3]; AC⃗ = [6, 9] = 3AB⃗ → collinear

Linear combinations

A linear combination combines scalar multiplication and addition:

2u − 3v, where u = [1, 4], v = [2, −1]

= [2, 8] − [6, −3] = [−4, 11]

Every 2D vector can be written as a linear combination of i and j: [5, −2] = 5i − 2j.

Properties

Property Rule
distributive (vectors) k(u + v) = ku + kv
distributive (scalars) (k + m)v = kv + mv
associative k(mv) = (km)v
identity 1v = v

Applications

  • Doubling a force doubles its effect: 2F.
  • Velocity × time (a scalar) = displacement.
  • Scaling a design or map.

Worked examples

Example 1: 4[−1, 3, 2] = [−4, 12, 8].

Example 2: Are [3, −6] and [−1, 2] parallel? [3, −6] = −3[−1, 2] → yes.

Example 3: Find k if [k, 6] is parallel to [2, 3]. k/2 = 6/3 → k = 4.

Common mistakes

  • Multiplying only one component.
  • Forgetting that negative k reverses direction.
  • Thinking |kv| = k|v| for negative k (use |k|).

Try it yourself

  1. −3[4, −2]
  2. |2v| if v = [3, 4]
  3. Are [5, 10] and [2, 4] parallel?
  4. u = [2, 0], v = [1, 3]. Find 3u + 2v.
  5. Are A(0, 0), B(2, 5), C(6, 15) collinear?

Answers: 1) [−12, 6] 2) 10 3) yes 4) [8, 6] 5) yes

Quick recap

  • Scalar multiplication: k[x, y] = [kx, ky]; negative k gives the opposite direction.
  • Parallel vectors are scalar multiples; use this to test collinear points.
  • Linear combinations mix scalar multiples and sums.