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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 Video by Zeeshan Zamurred Vectors

Edexcel A-Level Maths: Vectors (Part 3)Watch the full walkthrough 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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