Unit vectors out of the axes

Question

I'm having a difficulty to understand unit vectors other than those of the axes ($i, j, k$).

Say I have two vectors $U$ and $V$, somewhere in $\Bbb R^3$ in terms of $i, j, k$, and I want to know the projection of vector $U$ over vector $V$, if their coordinates are all non-zero.

As I see it, I should make:

$$\operatorname{proj}_vu = |u| \cos(\alpha)\;,$$ where $v$ is a unit vector. But how can I find an unit vector $V$ if its coordinates are non-zero? Its length won't ever be $1$.

I apologize for my lack of TEX knowledge, I'm sure it would have made this clearer. Tried to code it somewhere else and paste an url here, but it didn't work.

Answer 1

If you have a vector with non-zero coordinates, then it has non-zero length. Hence, you may always divide out by the length, to obtain a vector of length 1, that points in the direction of your original vector. We call this the unit vector

$$ \hat{u} = \frac {\vec{v} } { \left\| u \right \| }$$

For example, the vector $ ( 1, 2, 2) $ has length $\sqrt{ 1^2 + 2^2 + 2^2} = \sqrt{9} = 3$, thus the unit vector will be $ ( \frac {1}{3}, \frac {2}{3}, \frac {2}{3} ) $.