GCSE — mathsHigher1 min read

Vectors

Add, subtract, and use vectors in geometry for GCSE Maths. Represent vectors as column vectors and express paths.

Tutor AI Team2026-02-25T01:49:50.260Z

Vectors have both magnitude (size) and direction. In GCSE Maths, vectors are used to describe translations and prove geometric properties.

Core Concepts

Notation

Column vector: $\binom{3}{-2}$ (3 right, 2 down).
Named vector: $\vec{AB}$ or $\mathbf{a}$ .

Operations

Addition: $\binom{2}{3} + \binom{1}{-4} = \binom{3}{-1}$

Subtraction: $\binom{5}{2} - \binom{3}{1} = \binom{2}{1}$

Scalar multiplication: $3\binom{2}{-1} = \binom{6}{-3}$

Reverse Direction

$\vec{BA} = -\vec{AB}$

Expressing Paths

To go from A to C via B: $\vec{AC} = \vec{AB} + \vec{BC}$ .

Parallel Vectors

If $\vec{PQ} = k\vec{RS}$ for some scalar $k$ , the vectors are parallel.

Midpoints

If M is the midpoint of AB: $\vec{OM} = \vec{OA} + \frac{1}{2}\vec{AB}$ .

Worked Example: Example 1

Problem

$\vec{OA} = \mathbf{a}$ , $\vec{OB} = \mathbf{b}$ . Find $\vec{AB}$ .

$\vec{AB} = \vec{AO} + \vec{OB} = -\mathbf{a} + \mathbf{b} = \mathbf{b} - \mathbf{a}$

Solution

Worked Example: Example 2

Problem

M is the midpoint of AB. $\vec{OM} = \frac{1}{2}(\mathbf{a} + \mathbf{b})$ .

Solution

Practice Problems

1. $\vec{OA} = \binom{3}{4}$ , $\vec{OB} = \binom{7}{1}$ . Find $\vec{AB}$ .
1. Show that $\binom{6}{-4}$ is parallel to $\binom{3}{-2}$ .
1. M is the midpoint of PQ. $\vec{OP} = 2\mathbf{a}$ , $\vec{OQ} = 4\mathbf{b}$ . Find $\vec{OM}$ .

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Key Takeaways

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Vectors have magnitude and direction.
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$\vec{AB} = \vec{OB} - \vec{OA} = \mathbf{b} - \mathbf{a}$ .
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Parallel: one is a scalar multiple of the other.
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Midpoint: $\vec{OM} = \frac{1}{2}(\mathbf{a} + \mathbf{b})$ .

Tags:

#gcse#maths#geometry#vectors

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