Magnitude and direction of vectors

The idea

The magnitude of $a\mathbf{i} + b\mathbf{j}$ is $\sqrt{a^2 + b^2}$ . Its direction is an angle found with $\tan\theta = \frac{b}{a}$ and a sketch. A unit vector has magnitude $1$ : divide the vector by its magnitude.

Sketch the vector to see which quadrant it is in.
Magnitude by Pythagoras.
Angle with $\mathbf{i}$ : $\tan^{-1}\left|\frac{b}{a}\right|$ , then adjust using the sketch.

Watch out The magnitude is a length; the direction needs a separate angle.

See it worked

Worked example

Find the magnitude of $5\mathbf{i} - 12\mathbf{j}$ and the angle it makes with $\mathbf{i}$ .

$|5\mathbf{i} - 12\mathbf{j}| = \sqrt{25 + 144} = 13$ .
$\tan\theta = \frac{12}{5}$ , so $\theta = 67.4^\circ$ .
The vector points below $\mathbf{i}$ : $67.4^\circ$ below the $\mathbf{i}$ direction.

Answer $13$ ; $67.4^\circ$ below $\mathbf{i}$

Your turn

Question 1Non-calculator3 marks

Find $|6\mathbf{i} + 8\mathbf{j}|$ .

Question 2Non-calculator3 marks

Find the unit vector in the direction of $3\mathbf{i} - 4\mathbf{j}$ . Give the $\mathbf{i}$ and $\mathbf{j}$ components.

Question 3Non-calculator3 marks

Find the angle between $2\mathbf{i} + 2\mathbf{j}$ and the vector $\mathbf{i}$ , in degrees.

In past papers

63 official questions in 56 papers since 2018. The latest:

All A-level Maths past papers

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