Through M_1(1,-2,1) draw a line: 1^° parallel to the x…

Through $M_1(1,-2,1)$ draw a line: $1^\circ$ parallel to the $x$-axis; $2^\circ$ parallel to $\tfrac{x-2}{0}=\tfrac{y+2}{1}=\tfrac{z+3}{-3}$; $3^\circ$ parallel to $x-y+z-4=0,\ 2x+y-2z+5=0$; $4^\circ$ parallel to $x=2,\ y=3$.

Explanation of Method

Lines through M₁(1, −2, 1) parallel to four given directions.

Step 1: Solutions

	Direction	Line through $M_1$
$1^\circ$	$\{1,0,0\}$	$y=-2,\ z=1$
$2^\circ$	$\{0,1,-3\}$	$x=1,\ \tfrac{y+2}{1}=\tfrac{z-1}{-3}$
$3^\circ$	$\{1,4,3\}$ (Problem 224)	$\tfrac{x-1}1=\tfrac{y+2}4=\tfrac{z-1}3$
$4^\circ$	$\{0,0,1\}$	$x=1,\ y=-2$

Each direction is read off directly, or computed as $\vec n_1\times\vec n_2$ in case $3^\circ$. Each line then passes through $M_1$.

$1^\circ$ $2^\circ$ $3^\circ$ $4^\circ$ $M_1$

The four lines through $M_1(1,-2,1)$: $1^\circ$ parallel to $Ox$, $2^\circ$ direction $\{0,1,-3\}$, $3^\circ$ direction $\{1,4,3\}$, $4^\circ$ parallel to $Oz$.

Result

\[\begin{aligned} 1^\circ&\ y=-2,\ z=1\\ 2^\circ&\ x=1,\ \tfrac{y+2}{1}=\tfrac{z-1}{-3}\\ 3^\circ&\ \tfrac{x-1}1=\tfrac{y+2}4=\tfrac{z-1}3\\ 4^\circ&\ x=1,\ y=-2 \end{aligned}\]