Determine the angles with the coordinate axes of the lines…

Determine the angles with the coordinate axes of the lines: $1^\circ$ $x=2,\ y=1$; $2^\circ$ $x=1,\ z=-1$; $3^\circ$ $y=0,\ z=2$.

Explanation of Method

The angles that three lines, each parallel to a coordinate axis, make with the axes.

Step 1: Solution

Two fixed coordinates leave the third one free, so the line is parallel to that axis.

	Line	$\vec d$	$\alpha$	$\beta$	$\gamma$
$1^\circ$	$x=2,\ y=1$	$\{0,0,1\}$	$90^\circ$	$90^\circ$	$0^\circ$
$2^\circ$	$x=1,\ z=-1$	$\{0,1,0\}$	$90^\circ$	$0^\circ$	$90^\circ$
$3^\circ$	$y=0,\ z=2$	$\{1,0,0\}$	$0^\circ$	$90^\circ$	$90^\circ$

Taking the opposite direction, $0^\circ$ becomes $180^\circ$. A line determines its angles only up to this choice.

$1^\circ\ \parallel Oz$ $2^\circ\ \parallel Oy$ $3^\circ\ \parallel Ox$

$1^\circ$ $x=2,y=1$ is vertical; $2^\circ$ $x=1,z=-1$ is parallel to $Oy$; $3^\circ$ $y=0,z=2$ is parallel to $Ox$ (in the plane $xOz$).

Result

\[\begin{aligned} 1^\circ&\ \alpha=\beta=90^\circ,\ \gamma=0^\circ\\ 2^\circ&\ \alpha=\gamma=90^\circ,\ \beta=0^\circ\\ 3^\circ&\ \alpha=0^\circ,\ \beta=\gamma=90^\circ \end{aligned}\]