Finding Direction Cosines

What they are

The direction cosines of a line are the cosines of the angles it makes with the $x$-, $y$-, and $z$-axes. For any direction ratios $(a,b,c)$, the direction cosines are found by dividing each by the overall magnitude:

Worked example: a coordinate axis

Write the direction cosines of the $x$-axis.

Solution: The $x$-axis points entirely along $x$ with no $y$ or $z$ component, so its direction cosines are simply $(1,0,0)$.

Worked example: a line through two points

Find the direction cosines of the line joining $(1,0,0)$ and $(0,1,1)$.

Solution: The direction ratios are the difference of coordinates: $(0-1, 1-0, 1-0) = (-1,1,1)$. The magnitude is $\sqrt{1+1+1}=\sqrt{3}$, so the direction cosines are:

Worked example: from given angles

A line makes angles of $90^\circ$, $60^\circ$, and $30^\circ$ with the positive $x$-, $y$-, and $z$-axes respectively. Find its direction cosines.

Solution: Direction cosines are literally the cosines of those angles: