Use column vector notation, add and subtract vectors and multiply by a scalar, and use vectors to find resultants and to prove that lines are parallel or points are collinear (Higher tier).

What this dot point is asking

Vectors are a Higher-tier-only topic. WJEC asks you to use column vector notation, to add and subtract vectors and multiply them by a scalar, and to use vectors in geometric arguments, for example to find a resultant or to prove that two lines are parallel or three points are collinear. A vector has both magnitude and direction, and the power of the topic is that geometric relationships become algebra: a path around a shape is a sum of vectors. The proof questions reward expressing a journey in terms of given vectors and then drawing a conclusion.

Column vectors and notation

A column vector records movement as two components.

The magnitude (length) of $\binom{x}{y}$ is $\sqrt{x^2 + y^2}$ by Pythagoras, linking vectors back to the right-angled triangle.

Adding, subtracting and scaling

Vector arithmetic works component by component.

To add $\binom{x_1}{y_1} + \binom{x_2}{y_2} = \binom{x_1 + x_2}{y_1 + y_2}$, and to subtract do the same with minus signs. To multiply by a scalar $k$, multiply both components: $k\binom{x}{y} = \binom{kx}{ky}$. Geometrically, adding vectors places them nose to tail, so the resultant $\vec{AC} = \vec{AB} + \vec{BC}$ is the single vector from start to finish. Scaling by a positive number keeps the direction and changes the length; scaling by a negative number reverses it.

Vector paths

Any journey across a diagram is a sum of vectors.

This "path-finding" is the heart of vector questions: choose a route through known vectors, then add them with the correct signs.

Parallel lines and collinearity

Vectors prove geometric facts about direction.

Two vectors are parallel exactly when one is a scalar multiple of the other; if such parallel vectors also share a common point, the points lie on a straight line (collinear).

Why this matters

Vectors are a Higher-tier topic that rewards a different kind of thinking: turning a geometric diagram into vector algebra. The arithmetic (adding, subtracting and scaling column vectors) is routine, but the proof questions, expressing a journey in terms of given vectors and then arguing that vectors are parallel or points collinear, are demanding AO3 problems worth several marks. The skill of building a path with correct signs, and of recognising a scalar multiple as the signature of parallel lines, is what the marks reward.