Vectors (GCSE Maths)

GCSE Maths | Higher Tier

Key Facts

A vector has both magnitude (size) and direction.
A scalar has magnitude only (e.g. speed, temperature).
Column vectors are written as two numbers stacked: the top is the horizontal component, the bottom is the vertical component.
Vectors are equal if they have the same magnitude and direction, regardless of position.
A negative vector points in the opposite direction.

What Is a Vector?

A vector describes a movement from one point to another. It tells you how far to go and in which direction. Vectors are drawn as arrows: the length represents the magnitude and the arrowhead shows the direction. In GCSE maths, vectors are usually written as bold letters (like a) or with an arrow above (like AB with an arrow).

Column Vectors

A column vector like (3 over 2) means move 3 to the right and 2 up. A vector like (−1 over 4) means move 1 to the left and 4 up. The top number is always the horizontal movement (positive = right) and the bottom number is the vertical movement (positive = up).

Adding, Subtracting and Scalar Multiplication

To add vectors, add the corresponding components. To subtract, subtract them. To multiply by a scalar, multiply each component by that number. Multiplying by 2 doubles the length; multiplying by −1 reverses the direction.

Magnitude of a Vector

The magnitude (length) of a column vector (a over b) is found using Pythagoras: magnitude = the square root of (a squared + b squared). For example, the magnitude of (3 over 4) is the square root of (9 + 16) = the square root of 25 = 5.

Worked Examples

Example 1: Add the vectors (2 over 5) and (3 over −1)

Add horizontals: 2 + 3 = 5

Add verticals: 5 + (−1) = 4

Answer: (5 over 4)

Example 2: Find 3 times the vector (2 over −4)

Multiply each component by 3: (6 over −12)

Answer: (6 over −12)

Example 3: Find the magnitude of (5 over 12)

Magnitude = square root of (25 + 144) = square root of 169 = 13

Answer: 13 units

Common Mistakes

Mixing up the horizontal and vertical components.
Forgetting that subtracting a vector is the same as adding its negative.
Not simplifying route problems (e.g. going A to B via O: vector AO + vector OB).

Practice Questions

Add (4 over −3) and (−2 over 7).
Find 2 times (1 over 5) minus (3 over 2).
Find the magnitude of (−8 over 6).
If vector AB = (3 over 4), what is vector BA?
OA = a and OB = b. Find the vector AB in terms of a and b.