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12.2PureStretch

Vectors in 3D

A 3D vector has three components, written with the unit vectors i, j, k or as a column. Magnitude, addition and scalar multiplication all extend naturally from 2D.

24 min Vectors
Edexcel A-Level Maths: Vectors (Part 2)Curated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Write vectors in i, j, k or column form
  • Find the magnitude of a 3D vector
  • Add and scale 3D vectors
  • Find a unit vector
1

Components and magnitude

A vector ai+bj+cka\mathbf{i} + b\mathbf{j} + c\mathbf{k} has magnitude a2+b2+c2\sqrt{a^2 + b^2 + c^2}. The unit vectors i,j,k\mathbf{i}, \mathbf{j}, \mathbf{k} point along the x,y,zx, y, z axes.

∣ai+bj+ck∣=a2+b2+c2|a\mathbf{i} + b\mathbf{j} + c\mathbf{k}| = \sqrt{a^2 + b^2 + c^2}
Magnitude of a 3D vector.
Example 1Find the magnitude of v=2i−j+2k\mathbf{v} = 2\mathbf{i} - \mathbf{j} + 2\mathbf{k}.
122+(−1)2+22=4+1+4\sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4}.
2=9=3= \sqrt{9} = 3.
Answer
33
2

Operations and unit vectors

Add vectors component by component; multiply by a scalar to scale. A unit vector\textbf{unit vector} in the direction of v\mathbf{v} is 1∣v∣v\dfrac{1}{|\mathbf{v}|}\mathbf{v}.

Tip — To make a unit vector, divide each component by the magnitude.

Formula recap

∣v∣=a2+b2+c2|\mathbf{v}| = \sqrt{a^2 + b^2 + c^2}
Magnitude.
v^=1∣v∣v\hat{\mathbf{v}} = \frac{1}{|\mathbf{v}|}\mathbf{v}
Unit vector.

Common mistakes to avoid

Adding magnitudes instead of components.
Add component by component, then take the magnitude if needed.
Forgetting a component when squaring for magnitude.
Include all three squared components.

Key takeaways

  • Vectors: ai + bj + ck or column form.
  • |v| = √(a² + b² + c²).
  • Unit vector = v / |v|.

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 magnitude of ai+bj+cka\mathbf{i}+b\mathbf{j}+c\mathbf{k} is:

    • a2+b2+c2\sqrt{a^2+b^2+c^2}
    • a+b+ca+b+c
    • a2+b2\sqrt{a^2+b^2}
    • abcabc
    Show answer and explanation

    Answer: a2+b2+c2\sqrt{a^2+b^2+c^2}

    Square, add, root.

  2. The magnitude of 2i−j+2k2\mathbf{i}-\mathbf{j}+2\mathbf{k} is:

    • 33
    • 55
    • 7\sqrt{7}
    • 22
    Show answer and explanation

    Answer: 33

    √9.

  3. A unit vector in the direction of v is:

    • 1∣v∣v\frac{1}{|\mathbf{v}|}\mathbf{v}
    • ∣v∣v|\mathbf{v}|\mathbf{v}
    • v2\mathbf{v}^2
    • v2\frac{\mathbf{v}}{2}
    Show answer and explanation

    Answer: 1∣v∣v\frac{1}{|\mathbf{v}|}\mathbf{v}

    Divide by magnitude.

  4. To add two 3D vectors you:

    • add components
    • add magnitudes
    • multiply
    • take the average
    Show answer and explanation

    Answer: add components

    Component-wise.

  5. The unit vectors i, j, k point along the:

    • x, y, z axes
    • origin
    • a single axis
    • diagonal
    Show answer and explanation

    Answer: x, y, z axes

    Standard basis.

Next step

Check your understanding

Work through questions on Vectors in 3D. Use the feedback to decide what needs another look.

3 past-paper questions · 7 marks

Practise this topic

Related lessons

All 4 lessons

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