What is important to know about Vectors?

A vector has magnitude (size) and direction.

What is important to know about Vectors?

Column vector: (x, y) means x units right and y units up (negative = left/down).

What is important to know about Vectors?

Adding vectors: add corresponding components: (a, b) + (c, d) = (a+c, b+d).

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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.

Revising vectors the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Vectors GCSE” on the site for the complete guide.

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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

OM = OA + AM
AM = ½ AB = ½(b − a)
OM = a + ½(b − a) = a + ½b − ½a = ½a + ½b
OM = ½(a + b)

AnswerOM = ½(a + b)

Example 2

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

Working

2a = (6, −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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