# Finding the Unit Vector of a Vector

Every nonzero vector has a corresponding *unit vector,* which has the same direction as that vector but a magnitude of 1. To find the unit vector **u** of the vector

you divide that vector by its magnitude as follows:

Note that this formula uses scalar multiplication, because the numerator is a vector and the denominator is a scalar.

A scalar is just a fancy word for a real number. The name arises because a scalar *scales *a vector — that is, it changes the scale of a vector. For example, the real number 2 scales the vector **v** by a factor of 2 so that 2**v** is twice as long as **v**.

As you may guess from its name, the unit vector is a *vector.*

For example, to find the unit vector **u** of the vector

you first calculate its magnitude |**q**|:

Now use the previous formula to calculate the unit vector:

You can check that the magnitude of resulting vector **u** really is 1 as follows: