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 Video by Zeeshan Zamurred Vectors

Edexcel A-Level Maths: Vectors (Part 2)Watch the full walkthrough 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

Components and magnitude

A vector $a i + b j + c k$ has magnitude $a^{2} + b^{2} + c^{2}$ . The unit vectors $i, j, k$ point along the $x, y, z$ axes.

∣ a i + b j + c k ∣ = a^{2} + b^{2} + c^{2}

Magnitude of a 3D vector.

2^{2} + (- 1)^{2} + 2^{2} = 4 + 1 + 4

= 9 = 3

Answer

3

Operations and unit vectors

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

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

Formula recap

∣ v ∣ = a^{2} + b^{2} + c^{2}

Magnitude.

\hat{v} = \frac{1}{∣ v ∣} 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

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