183_notes:examples:unitvector [Projects & Practices in Physics]

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===== Example: Calculating a unit vector =====

You will find it useful to be able to calculate the unit vector of a given vector. For example, the [[http://en.wikipedia.org/wiki/Drag_(physics)#Types_of_drag|force due to air resistance]] is defined in terms of the velocity unit vector. 

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

==== Solution ====

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.

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