Vectors for GCSE Maths

Vectors in GCSE Maths describe a movement with both a size and a direction. You write them as column vectors, with the move across on top and the move up underneath, and you add, subtract and multiply them by numbers. At Higher tier you also use vectors to prove geometric facts, such as two lines being parallel.

What is a vector?

A vector has a magnitude (size) and a direction. A scalar has only a size: 5 m is a scalar, but 5 m north is a vector. In GCSE Maths, vectors usually describe movements, like the translations you meet in transformations. AQA's GCSE specification asks you to describe translations as 2D vectors, and to apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors.

How vectors are written

• As a column vector: the top number is the move to the right (negative means left), the bottom number is the move up (negative means down).
• As a bold letter, such as a. Handwritten, it is underlined: a̲.
• As two letters with an arrow, such as AB with an arrow above, meaning the vector from A to B.

This vector means 3 to the right and 2 down.

Adding and subtracting column vectors

Add or subtract the top numbers, then the bottom numbers. With a = (3, −2) and b = (−1, 5):

On a diagram, adding means going along one vector and then the next, nose to tail. The result, called the resultant, goes straight from the start of the first to the end of the second.

Subtracting a vector is adding its negative. The vector −b has the same length as b but points the opposite way.

Multiplying by a scalar

Multiply both numbers by the scalar. 3a = (9, −6) is three times as long as a, in the same direction. A negative scalar reverses the direction: −a = (−3, 2). Combining operations:

Translations and magnitude

A translation by a column vector moves every point the same way. Translating the point (2, 5) by the vector (−4, 1) gives (2 − 4, 5 + 1) = (−2, 6).

The length (magnitude) of a vector comes from Pythagoras, because its two numbers are the sides of a right-angled triangle. The magnitude of (5, −12) is √(5² + 12²) = √169 = 13. The vectors calculator works out magnitudes and resultants, and shows them on a grid.

Vector geometry proofs

AQA's GCSE specification also asks you to use vectors to construct geometric arguments and proofs. The method is always the same: write each vector you need as a route through vectors you already know.

The route rule

To find a vector from one point to another, follow any path along known vectors. Going with a vector, add it; going against it, subtract it. In triangle OAB, with OA = a and OB = b, the route from A to B goes back along OA and then along OB:

Example: proving lines are parallel

In triangle OAB, OA = a and OB = b. X is the midpoint of OA and Y is the midpoint of OB. Prove that XY is parallel to AB.

1. OX = ½a and OY = ½b, because X and Y are midpoints.
2. XY = XO + OY = −½a + ½b = ½(b − a).
3. AB = b − a, from the route rule.
4. So XY = ½ AB. One is a multiple of the other, so XY is parallel to AB, and half its length.

Example: a midpoint

With the same triangle, M is the midpoint of AB. Find OM. Go from O to A, then halfway along AB: OM = a + ½(b − a) = a + ½b − ½a = ½(a + b).

To show three points are on a straight line (collinear), show that the vectors between them are multiples of each other and share a point. For example, if PQ = 2PR, then P, Q and R lie on one line through P.

Common mistakes

• Swapping the numbers. The top number is across, the bottom is up, the same order as coordinates.
• Writing a vector as a fraction. There is no line between the numbers in a column vector.
• Losing a minus sign on the route. Going against a vector subtracts it: from A to O is −a.
• Saying "parallel" without the reason. State that one vector is a multiple of the other.
• Forgetting the shared point for collinear. Parallel vectors alone don't prove three points are on one line.

To practise vectors with other GCSE topics in a short timed paper, try today's GCSE Maths Higher or GCSE Maths Foundation paper.

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