You will find it useful to be able to calculate the unit vector of a given vector. For example, the force due to air resistance is defined in terms of the velocity unit vector.

Determine the unit vector for the velocity vector, $\vec{v} = \langle 2, -1, 3 \rangle \dfrac{m}{s}$.

The unit vector is defined as the ratio of the vector itself to its magnitude. So, we can compute,

$$\hat{v} = \dfrac{\vec{v}}{|\vec{v}|} = \dfrac{\langle 2, -1, 3 \rangle \dfrac{m}{s}}{\sqrt{2^2+(-1)^2+3^2} \dfrac{m}{s}} = \dfrac{\langle 2, -1, 3 \rangle \dfrac{m}{s}}{\sqrt{14} \dfrac{m}{s}} = \langle 0.53, -.27, .80 \rangle$$

Notice that the unit vector has no units of its own. It simply represents the direction of this velocity vector. We can further check that it is indeed a unit vector by taking it's magnitude, which should be 1.

$$|\hat{v}| = \sqrt{0.53^+(-.27)^2+.80^2} = 1$$