We can find the angle between two nonzero vectors using the dot product:

`vec u * vec v = |vec u||vec v|cos theta qquad => qquad cos theta = (vec u * vec v)/(|vec u||vec v|)`.

Often, we want to determine the angle that a certain vector makes with the coordinate axes. To do this, we just put `hat i`, `hat j`, and `hat k` in the place of `vec v` one at a time and solve for the angle. Since this operation is so common, it is worthwhile to work out specific equations.

Let’s start with the *x*-axis. If we have a vector `vec u` in component form, then

`cos theta = (vec u * hat i)/(|vec u||hat i|) = ([u_1,u_2,u_3] * [1,0,0])/(|vec u|(1)) = u_1/|vec u|`.

When we dot `vec u` with one of the standard basis vectors, we are effectively *choosing* one of its components. Simplifying the equation for the other two axes is just as easy. To avoid confusion, we use three different Greek letters to represent the three angles:

Axis | Basis | Cosine | Value |
---|---|---|---|

x |
`hat i` | `cos alpha` | `u_1//|vec u|` |

y |
`hat j` | `cos beta` | `u_2//|vec u|` |

z |
`hat k` | `cos gamma` | `u_3//|vec u|` |