Vectors (Years 10–11) — Maths

Tell someone "walk 3 blocks east then 2 blocks north" and you have just given them a vector: how far and in which direction to move. Video games, sat-navs and physics all run on vectors. We write one as two numbers — the first is movement right, the second is movement up (negatives mean left or down).

Why this matters: vectors are the language of movement and force. They connect straight to translations in geometry, to velocity in physics, and to the coordinate geometry you meet later. Mastering the simple part-by-part arithmetic now makes all of that easy.

The vector (3, 2): move 3 right, then 2 up

Reading a vector

Read the top number as an across move and the bottom number as an up move. A negative flips the direction of that part.

(3, 2) means "3 right, 2 up". (−4, 1) means "4 left, 1 up". (0, −5) means "0 across, 5 down".

The vector (−4, 1): a negative first part means move left.

Adding and subtracting

To add vectors, add the first numbers together and the second numbers together — it is exactly like doing one move and then the other, so the totals stack up. Subtract in the same part-by-part way.

(3, 2) + (1, 4) = (4, 6). (5, 3) − (2, 1) = (3, 2). Think of walking two journeys back to back: the combined trip is the sum.

Add tip-to-tail: (3, 2) then (1, 4) lands at the resultant (4, 6).

Multiplying by a number (a scalar)

Multiply both parts by the number. This stretches the vector without turning it — like taking the same step twice or three times over. A negative scalar reverses the direction.

(3, 2) × 2 = (6, 4). (6, 4) × ½ = (3, 2). The negative of (3, 5) is (−3, −5) — same length, opposite way.

Multiplying by a scalar stretches the vector without turning it: 2v is twice as long.

Operation	Rule	Example
Add	Add each part	(3, 2) + (1, 4) = (4, 6)
Subtract	Subtract each part	(5, 3) − (2, 1) = (3, 2)
Multiply by scalar	Multiply both parts	(2, 4) × 3 = (6, 12)
Negative	Flip both signs	−(3, 5) = (−3, −5)

Common mistake: mixing up the rows. Keep the parts lined up — first with first, second with second. Never add an across number to an up number, and when subtracting, watch the signs: (4, 1) − (1, 3) = (3, −2), not (3, 2).

In a nutshell: a vector is (right, up); add or subtract part-by-part; multiply both parts by a scalar to stretch it; and the negative points the opposite way.