\( \newcommand\mycolv[1]{\begin{pmatrix}#1\end{pmatrix}} \)
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Magnitude and Direction

Calculate the magnitude and direction of a vector and convert between forms.

Menu Connect Vectors Level 1 Level 2 Level 3 Exam Help More

You can draw on a coordinate grid to help you answer some of the questions below.


Find the length of the vector:

\( \mycolv{3\\4} \)


Find the magnitude of:

\( \mycolv{12\\5} \)


Find the length of the vector:

\( \mycolv{8\\6} \)


Find the magnitude of

\( \mycolv{12\\16} \)

(to 3 significant figures)


Find the magnitude of

\( \mycolv{-13\\14} \)

(to 3 significant figures)


Find the magnitude of

\( \mycolv{12\\-18} \)

(to 3 significant figures)


Find the magnitude of

\( \mycolv{-9\\20} \)

(to 3 significant figures)


Find the magnitude of

\( \mycolv{6\\12} \)

(to 3 significant figures)


Find the magnitude of

\( \mycolv{-13\\-19} \)

(to 3 significant figures)


This is Vectors - Magnitude and Direction level 1. You can also try:
Vector Connectors Level 2 Level 3


Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

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Description of Levels


Vector Connectors - A basic set of exercises on vectors which could be done before attempting the following.

Vectors - Addition and subtraction of vectors and multiplication of a vector by a scalar.

Level 1 - Magnitude of a vector from a column vector

Level 2 - Direction of a vector from a column vector

Level 3 - Finding the column vector from magnitude-direction form

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

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Curriculum Reference

See the National Curriculum page for links to related online activities and resources.


This set of exercises is to provide you with practice calculating the magnitude and direction of a vector and converting between component form and magnitude/direction form.

The magnitude of a vector is the length of the vector. A length cannot be negative. If the horizontal and vertical components are known it can be calculated using Pythagoras' theorem.

For example consider the vector:

$$ \mycolv{9\\12} $$

The magnitude is:

$$ \sqrt{9^2 + 12^2} = 15$$

The direction of the vector can be calculated using trigonometry. The angle the vector makes with the horizontal is the inverse tan of the vertical component divide by the horizontal component.

$$ \tan^{-1} \frac{12}{9} = 53.1^o$$

If the magnitude and direction are given, as in the diagram below, trigonometry can be used to calculate the components for the column vector.

Diagram 1 $$ x = 18 \cos 61^o = 8.73$$ $$ y = 18 \sin 61^o = 15.7$$

So the column vector, with components expressed to 3 significant figures, is:

$$ \mycolv{8.73\\15.7} $$

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