Vector Magnitude

Vector magnitude: how to find it

The magnitude (or length) of a vector tells you how long the vector is. For a vector written in components, such as (\langle a, b \rangle) or ((a, b)), use the Pythagorean idea:

[ |\vec v| = \sqrt{a^2 + b^2} ]

For a vector in three components, use

[ |\vec v| = \sqrt{a^2 + b^2 + c^2}. ]

Steps to follow

1. Identify the components of the vector.
2. Square each component.
3. Add the squares.
4. Take the square root of the result.
5. Simplify the radical if possible.

Example method

If the vector is (\langle 3, 4 \rangle), then

[ |\vec v| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. ]

Check your answer

A good check is to see whether the length makes sense. For integer components, the result is often a whole number or a simplified radical. Also, make sure you used every component exactly once and squared them before adding. If the final answer can be simplified, write it in simplest exact form.