11.3PureCore

Magnitude and Direction

The magnitude of a vector is its length, found with Pythagoras. Its direction is the angle it makes, found with trigonometry. Together they let you describe any vector fully and build unit vectors.

25 min Vectors

Edexcel AS Level Maths: 11.3 Magnitude and DirectionCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Find the magnitude of a vector
Find the direction (angle) of a vector
Work out a unit vector in a given direction
Move between magnitude–direction and component form

Magnitude

The magnitude $|\mathbf{a}|$ of $\begin{pmatrix} x \\ y \end{pmatrix}$ is the length of the arrow — straight from Pythagoras.

\left| \begin{pmatrix} x \\ y \end{pmatrix} \right| = \sqrt{x^{2} + y^{2}}

The components are the legs of a right-angled triangle.

Example 1Find the magnitude of

\begin{pmatrix} 3 \\ 4 \end{pmatrix}

\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}

Answer

5

Direction

The direction is the angle the vector makes, found with $\tan$ using the components. Always draw a quick sketch to get the correct angle from the right reference direction.

\theta = \tan^{-1}\!\left( \dfrac{y}{x} \right)

Use a sketch to place the angle correctly.

Unit vectors

A $\textbf{unit vector}$ has magnitude $1$ . To find the unit vector in the direction of $\mathbf{a}$ , divide $\mathbf{a}$ by its own magnitude.

\hat{\mathbf{a}} = \dfrac{1}{|\mathbf{a}|}\,\mathbf{a}

Same direction, length scaled to 1.

Example 1Find the unit vector in the direction of

\begin{pmatrix} 3 \\ 4 \end{pmatrix}

1Magnitude is

5

2Divide each component by 5.

Answer

\begin{pmatrix} 3/5 \\ 4/5 \end{pmatrix}

Tip — Check a unit vector by squaring its components — they should add to 1.

Formula recap

|\mathbf{a}| = \sqrt{x^{2} + y^{2}}

Magnitude (Pythagoras).

\theta = \tan^{-1}\!\left(\tfrac{y}{x}\right)

Direction angle.

\hat{\mathbf{a}} = \dfrac{\mathbf{a}}{|\mathbf{a}|}

Unit vector.

Common mistakes to avoid

Forgetting the square root when finding magnitude.

|a| = √(x² + y²), not x² + y².

Reading the direction angle off without a sketch.

A quick diagram ensures the angle is measured from the right direction.

Key takeaways

Magnitude |a| = √(x² + y²).
Direction comes from tan⁻¹(y/x) — sketch to place the angle.
A unit vector is a/|a| — same direction, length 1.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Check your understanding

Work through questions on Magnitude and Direction. Use the feedback to decide what needs another look.

2 past-paper questions · 4 marks

Practise this topic

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