Write the projections of tfracx+12=tfrac y1=(z-1)/(-1) onto…

Write the projections of $\tfrac{x+1}2=\tfrac y1=\tfrac{z-1}{-1}$ onto the coordinate planes.

Explanation of Method

The projections of a line onto the three coordinate planes.

Step 1: Solution

Equate the fractions in pairs. Each pair omits one variable and is a plane perpendicular to one coordinate plane; its intersection with that coordinate plane is the projection.

Onto	From	Projection
$xOy$	$\tfrac{x+1}2=\tfrac y1$	$x-2y+1=0,\ z=0$
$yOz$	$\tfrac y1=\tfrac{z-1}{-1}$	$y+z-1=0,\ x=0$
$xOz$	$\tfrac{x+1}2=\tfrac{z-1}{-1}$	$x+2z-1=0,\ y=0$

The directions of the projections are $\{2,1,0\}$, $\{0,1,-1\}$ and $\{2,0,-1\}$: the direction of the line with one component dropped.

$xOy$ $yOz$ $xOz$

The line (black) and its projections: onto $xOy$ (blue) $x-2y+1=0$, onto $yOz$ (red) $y+z=1$, onto $xOz$ (teal) $x+2z=1$.

Result

\[\begin{aligned} xOy&:\ x-2y+1=0,\ z=0\\ yOz&:\ y+z-1=0,\ x=0\\ xOz&:\ x+2z-1=0,\ y=0 \end{aligned}\]