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Geometry & Measures · Higher

Vectors

Vectors represent quantities with both magnitude and direction. In GCSE, vectors are written as column vectors or using bold/underlined letters, and are used to prove geometric results.

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Key facts to remember

  • 1A vector has magnitude (size) and direction.
  • 2Column vector: (x, y) means x units right and y units up (negative = left/down).
  • 3Adding vectors: add corresponding components: (a, b) + (c, d) = (a+c, b+d).
  • 4Multiplying by a scalar: k(a, b) = (ka, kb).
  • 5The negative of a vector reverses its direction: −a goes the opposite way to a.
  • 6Parallel vectors are scalar multiples of each other.
  • 7To prove three points are collinear: show one vector is a scalar multiple of another, sharing a common point.
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Formulas

Magnitude of vector
|v| = √(x² + y²)

For column vector (x, y)

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Worked examples

Example 1

OA = a and OB = b. M is the midpoint of AB. Find the vector OM.

Working

  1. OM = OA + AM
  2. AM = ½ AB = ½(b − a)
  3. OM = a + ½(b − a) = a + ½b − ½a = ½a + ½b
  4. OM = ½(a + b)
AnswerOM = ½(a + b)
Example 2

Given a = (3, −1) and b = (−2, 4), find 2a − b.

Working

  1. 2a = (6, −2)
  2. 2a − b = (6 − (−2), −2 − 4) = (8, −6)
Answer(8, −6)
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Common mistakes

✗Confusing the direction of a vector: AB = B − A, not A − B.
✗Forgetting to use the midpoint formula correctly when finding vectors through midpoints.
✗Not simplifying vector expressions fully.
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Exam tips

✓Always write vector paths clearly: e.g. to go from A to C via B: AC = AB + BC.
✓To prove collinearity: show that two vectors sharing a point are parallel (one is a multiple of the other).

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