Use vector notation; add, subtract and multiply vectors by a scalar; and use vectors to construct geometric arguments and proofs (proof at Higher tier).

What this dot point is asking

OCR references G24 and G25 cover vectors: the notation, adding, subtracting and scaling vectors, and at Higher tier using them to build geometric arguments and proofs. A vector has both magnitude and direction, which makes it the natural tool for translations and for describing paths around a figure. Vector geometry, especially proof, is a demanding Higher-tier topic that tests AO2 reasoning and is a reliable source of the hardest marks on the paper.

Vector notation and arithmetic

A vector describes a movement, not a position.

To add or subtract, combine components: $\begin{pmatrix} 5 \\ 2 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$. To scale, multiply each component: $3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}$. The magnitude of a column vector is found with Pythagoras: $\left|\begin{pmatrix} 3 \\ 4 \end{pmatrix}\right| = \sqrt{3^2 + 4^2} = 5$.

Vectors around a figure

Vector geometry adds vectors nose to tail along a path.

So in a triangle $OAB$ with $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$, the side $\overrightarrow{AB}$ is $\mathbf{b} - \mathbf{a}$. Midpoints and points dividing a line in a ratio are reached by scaling: the midpoint $M$ of $AB$ gives $\overrightarrow{AM} = \tfrac{1}{2}(\mathbf{b} - \mathbf{a})$. Drawing the figure and marking each known vector keeps the route clear.

When a point divides a line in a given ratio rather than at the midpoint, scale by the matching fraction. If $P$ divides $AB$ so that $AP : PB = 1 : 2$, then $\overrightarrow{AP} = \tfrac{1}{3}\overrightarrow{AB} = \tfrac{1}{3}(\mathbf{b} - \mathbf{a})$, because $P$ is one third of the way along. Then $\overrightarrow{OP} = \mathbf{a} + \tfrac{1}{3}(\mathbf{b} - \mathbf{a})$, which simplifies to $\tfrac{2}{3}\mathbf{a} + \tfrac{1}{3}\mathbf{b}$. Setting up the ratio fraction correctly is the step that most often goes wrong, so always check that the fractions of the two parts add to one.

Vector proof (Higher)

The power of vectors is proving geometric facts.

Why vectors matter

Vectors connect geometry to algebra and to the transformations topic, and they give a clean way to prove parallelism, collinearity and ratios that would be awkward with angles alone. OCR's vector-proof questions are among the most discriminating on the Higher paper, and they reward a clear path argument with each step justified. Recognising that "parallel" means "a scalar multiple" is the single most useful insight.