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12.3PureStretch

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
1

Position and direction

The vector from AA to BB is AB→=b−a\overrightarrow{AB} = \mathbf{b} - \mathbf{a}. Two vectors are parallel\textbf{parallel} if one is a scalar multiple of the other.

AB→=b−a\overrightarrow{AB} = \mathbf{b} - \mathbf{a}
Displacement from position vectors.
Example 1Are u=2i+4j\mathbf{u} = 2\mathbf{i} + 4\mathbf{j} and v=i+2j\mathbf{v} = \mathbf{i} + 2\mathbf{j} parallel?
1u=2v\mathbf{u} = 2\mathbf{v}.
2A scalar multiple, so yes.
Answer
Parallel (u = 2v).
2

Angle between vectors

The angle θ\theta between a\mathbf{a} and b\mathbf{b} satisfies cos⁡θ=a⋅b∣a∣∣b∣\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}, where the dot product a⋅b=a1b1+a2b2+a3b3\mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3.

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

Formula recap

AB→=b−a\overrightarrow{AB} = \mathbf{b} - \mathbf{a}
Displacement.
cos⁡θ=a⋅b∣a∣∣b∣\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}
Angle between vectors.
a⋅b=0⇒perpendicular\mathbf{a}\cdot\mathbf{b} = 0 \Rightarrow \text{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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The displacement AB→\overrightarrow{AB} equals:

    • b−a\mathbf{b} - \mathbf{a}
    • a−b\mathbf{a} - \mathbf{b}
    • a+b\mathbf{a} + \mathbf{b}
    • a⋅b\mathbf{a}\cdot\mathbf{b}
    Show answer and explanation

    Answer: b−a\mathbf{b} - \mathbf{a}

    End minus start.

  2. Two vectors are parallel if:

    • one is a scalar multiple of the other
    • their dot product is zero
    • they are equal
    • they sum to zero
    Show answer and explanation

    Answer: one is a scalar multiple of the other

    Same direction.

  3. The angle between vectors uses:

    • cos⁡θ=a⋅b∣a∣∣b∣\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}
    • sin⁡θ=a⋅b\sin\theta = \mathbf{a}\cdot\mathbf{b}
    • tan⁡θ=∣a∣\tan\theta = |\mathbf{a}|
    • the sine rule
    Show answer and explanation

    Answer: cos⁡θ=a⋅b∣a∣∣b∣\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}

    Dot product formula.

  4. If a⋅b=0\mathbf{a}\cdot\mathbf{b} = 0 the vectors are:

    • perpendicular
    • parallel
    • equal
    • opposite
    Show answer and explanation

    Answer: perpendicular

    cos θ = 0.

  5. Are 2i+4j2\mathbf{i}+4\mathbf{j} and i+2j\mathbf{i}+2\mathbf{j} parallel?

    • Yes
    • No
    • Only in 2D
    • Cannot tell
    Show answer and explanation

    Answer: Yes

    First = 2× second.

Next step

Check your understanding

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

4 past-paper questions · 9 marks

Practise this topic

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