Vectors summary

Key learning points

• A column vector $\begin{pmatrix}x\\y\end{pmatrix}$ means $x$ right (negative = left) and $y$ up (negative = down).
• Arrow notation $\overrightarrow{AB}$ is the displacement from A to B; $\overrightarrow{BA}=-\overrightarrow{AB}$.
• Bold/underlined letters name the same displacement in column form.
• Add/subtract component-wise; multiply by a scalar to stretch or reverse.
• Magnitude $|\mathbf{a}|=\sqrt{x^2+y^2}$; geometry links use midpoints and parallelograms.

Worked example 1 — Addition

$\mathbf{a}=\begin{pmatrix}2\\1\end{pmatrix}$, $\mathbf{b}=\begin{pmatrix}3\\-4\end{pmatrix}$. $\mathbf{a}+\mathbf{b}$?

Working

Add components.

Worked example 2 — Midpoint vector

M is midpoint of $AB$. Express $\overrightarrow{OM}$ if $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$.

Working

Position vector of a midpoint.

Worked example 3 — Scalar multiple

$3\begin{pmatrix}2\\-1\end{pmatrix}$.

Working

Multiply each component.

Worked example 4 — Magnitude

Find $\left|\begin{pmatrix}3\\4\end{pmatrix}\right|$.

Working

$\sqrt{3^2+4^2}$.

Worked example 5 — Reverse

If $\overrightarrow{AB}=\mathbf{v}$, then $\overrightarrow{BA}$?

Working

Opposite direction.

Worked example 6 — Successive translations

A point is translated by $\begin{pmatrix}4\\1\end{pmatrix}$ then $\begin{pmatrix}-1\\3\end{pmatrix}$. Find one equivalent translation.

Working

Order does not matter for free-vector addition — both slides land on the same image.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.