Find out how long the vector is. ![](https://media1.shmoop.com/images/calculus/calc_fn_vect_narr_latek_73.png)
This vector is too short. We want it to have length 1, but it only has length . P![](https://media1.shmoop.com/images/calculus/calc_fn_vect_narr_graphik_47.png) If we multiply the vector by 2, we'll get a new vector with magnitude 1. Let
![](https://media1.shmoop.com/images/calculus/calc_fn_vect_narr_latek_75.png) The magnitude of v is ![](https://media1.shmoop.com/images/calculus/calc_fn_vect_narr_latek_76.png)
Therefore v is a unit vector, which is what we wanted. Remember your fraction division: multiplying by 2 is the same thing as dividing by . ![](https://media1.shmoop.com/images/calculus/calc_fn_vect_narr_latek_78.png)
Since multiplying by 2 is the same thing as dividing by , we could also write the vector as ![](https://media1.shmoop.com/images/calculus/calc_fn_vect_narr_latek_82.png)
which happens to be the same thing as ![](https://media1.shmoop.com/images/calculus/calc_fn_vect_narr_latek_83.png)
If we're given a vector v and asked to normalize it, find the vector ![](https://media1.shmoop.com/images/calculus/calc_fn_vect_narr_latek_84.png)
This is guaranteed to be a unit vector pointing in the same direction as v (unless v is 0, but that's not a vector). What direction does <0, 0> point? It doesn't. |