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Solving Geometric Problems

Vectors describe geometry algebraically. Position vectors, parallel conditions and the angle between vectors let you prove results and solve 3D problems.

24 min Vectors

Edexcel A-Level Maths: Vectors (Part 3)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Use position vectors
Test for parallel vectors
Find the angle between vectors
Solve geometric problems in 3D

Position and direction

The vector from $A$ to $B$ is $A B = b - a$ . Two vectors are $parallel$ if one is a scalar multiple of the other.

A B = b - a

Displacement from position vectors.

u = 2 v

2A scalar multiple, so yes.

AnswerParallel (u = 2v).

Angle between vectors

The angle $θ$ between $a$ and $b$ satisfies $cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}$ , where the dot product $a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$ .

Tip — A dot product of zero means the vectors are perpendicular.

Formula recap

A B = b - a

Displacement.

cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}

Angle between vectors.

a \cdot b = 0 \Rightarrow perpendicular

Perpendicularity.

Common mistakes to avoid

Writing AB = a − b.

AB = b − a (end minus start).

Forgetting to divide by the magnitudes in the angle formula.

cos θ = (a·b)/(|a||b|).

Key takeaways

AB = b − a.
Parallel ⇔ one is a scalar multiple of the other.
cos θ = (a·b)/(|a||b|); a·b = 0 ⇒ perpendicular.

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Real past-paper questions on Solving Geometric Problems, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.

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Exam questions

4 real exam questions on Solving Geometric Problems

9 marks · 2 papers — pulled from past papers and marked mark‑by‑mark, so what you miss feeds straight into your weak‑topic list.

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Solving Geometric Problems

Position and direction

Angle between vectors

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4 real exam questions on Solving Geometric Problems

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Solving Geometric Problems

Position and direction

Angle between vectors

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4 real exam questions on Solving Geometric Problems

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