11.5PureStretch

Solving Geometric Problems

Vectors give slick proofs in geometry. Using scalar multiples and the rule AB = b − a, you can prove lines are parallel, points are collinear, and split lines in given ratios — all algebraically.

30 min Vectors

Edexcel AS Level Maths: 11.5 Solving Geometric Problems (Vectors)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Prove two vectors (lines) are parallel
Show three points are collinear
Use ratios to find points along a line
Express unknown vectors in terms of given ones

Parallel and collinear

If $\overrightarrow{AB} = k\,\overrightarrow{CD}$ the lines are $\textbf{parallel}$ . If two vectors share a common point AND one is a scalar multiple of the other, the three points are $\textbf{collinear}$ (on one straight line).

\overrightarrow{AB} = k\,\overrightarrow{BC} \;\Rightarrow\; A, B, C \text{ collinear}

Parallel + shared point B ⟶ a single straight line.

Tip — Collinear = parallel vectors that share a point. Always state the shared point in your conclusion.

Working with ratios

If a point $P$ divides $AB$ in the ratio $m:n$ , you can write its position vector as a weighted combination. A point that splits $AB$ in ratio $1:1$ is just the midpoint $\tfrac{1}{2}(\mathbf{a} + \mathbf{b})$ .

\text{midpoint of } AB = \tfrac{1}{2}(\mathbf{a} + \mathbf{b})

A ratio of

1:1

averages the position vectors.

Example 1Find the position vector of the midpoint of

A(2,1)

and

B(6,5)

1Average:

\tfrac{1}{2}\left(\begin{pmatrix}2\\1\end{pmatrix} + \begin{pmatrix}6\\5\end{pmatrix}\right)

Answer

\begin{pmatrix} 4 \\ 3 \end{pmatrix}

Expressing unknown vectors

In a labelled diagram, build the vector you want by travelling along known vectors (nose-to-tail), using $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ as needed. Then compare scalar multiples to draw conclusions.

Tip — Always travel along edges you know. There is usually more than one valid route — pick the simplest.

Formula recap

\overrightarrow{AB} = k\,\overrightarrow{CD} \Rightarrow \text{parallel}

Parallel test.

\text{parallel} + \text{shared point} \Rightarrow \text{collinear}

Collinearity.

\text{midpoint} = \tfrac{1}{2}(\mathbf{a} + \mathbf{b})

Midpoint of AB.

Common mistakes to avoid

Claiming points are collinear just because vectors are parallel.

They must also share a common point to be collinear.

Getting the direction of travel wrong when building a route.

Use AB = b − a and reverse a vector when you travel against it.

Key takeaways

Parallel: one vector is a scalar multiple of the other.
Collinear: parallel vectors that also share a point.
Midpoint of AB = ½(a + b); ratios give weighted combinations.
Build unknown vectors by travelling along known ones.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Solving Geometric Problems. Use the feedback to decide what needs another look.

5 past-paper questions · 13 marks

Practise this topic

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