Vectors Topic Review summary

Key learning points

• $\overrightarrow{AB}=\mathbf{b}-\mathbf{a}$.
• $|\mathbf{a}|=\sqrt{a_1^2+a_2^2+a_3^2}$; unit vector $\hat{\mathbf{a}}=\mathbf{a}/|\mathbf{a}|$.
• Scalar product $\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos\theta$; perpendicular ⇒ 0.
• Line $\mathbf{r}=\mathbf{a}+\lambda\mathbf{d}$; classify intersecting/parallel/skew.
• Angle between lines uses direction vectors in the scalar product.

Worked example 1 — Dot

Angle between $\mathbf{i}+2\mathbf{j}$ and $3\mathbf{i}-\mathbf{j}$.

Working

Use the scalar product formula.

Worked example 2 — Point on line

$\mathbf{r}=(1,0,2)+\lambda(1,1,0)$ at $\lambda=2$.

Working

Substitute $\lambda$.

Worked example 3 — \AB=\b-\a

$\\overrightarrow{AB}=\\mathbf{b}-\\mathbf{a}$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — |\a|=\a_1^2+a_2^2+a_3^2; unit vector \\\ma

$|\\mathbf{a}|=\\sqrt{a_1^2+a_2^2+a_3^2}$; unit vector $\\hat{\\mathbf{a}}=\\mathbf{a}/|\\mathbf{a}|$.

Working

Attempt without notes, then check the answers PDF.

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.