Vectors summary

Key learning points

• $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$ from position vectors.
• Magnitude $|\mathbf{a}|=\sqrt{x^2+y^2(+z^2)}$; unit vector $\mathbf{a}/|\mathbf{a}|$.
• Scalar product $\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta$.
• Perpendicular vectors have scalar product zero.

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 — Scalar multiple

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

Working

Multiply each component.

Worked example 3 — Magnitude

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

Working

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

Worked example 4 — Reverse

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

Working

Opposite direction.

Worked example 5 — Scalar product

$\mathbf{a}=\begin{pmatrix}2\\0\\3\end{pmatrix}$, $\mathbf{b}=\begin{pmatrix}1\\-4\\2\end{pmatrix}$. Find $\mathbf{a}\cdot\mathbf{b}$.

Working

Multiply corresponding components and add.

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.