Solution

The vector from the origin $O$ to point $P=(4,-5,1)$ is

$$OP = 4\mathbf a_x -5\mathbf a_y + \mathbf a_z$$

But this is probably not a unit vector (in fact, it should be easy to tell that it definitely isn't). Recall that a unit vector is a vector of length 1 (that is, ``unit length"). So we just need to scale this vector by dividing it by its length to get a unit vector in its same direction.

The length of this vector is calculated using the 3-dimensional version of the Pythagorean theorem:

$$|OP| = \sqrt{4^2 + (-5)^2 + 1^2} = \sqrt{21}$$

Hence the unit vector is

$$\mathbf a_{OP} = \frac{OP}{|OP|} = \frac 4{\sqrt{21}}\,\bold a_x -\frac 5{\sqrt{21}}\,\bold a_y + \frac 1{\sqrt{21}}\,\bold a_z$$

Answer