Write the equations of the line through M_1(1,-1,1) parallel…

Write the equations of the line through $M_1(1,-1,1)$ parallel to $\vec p=\{1,2,3\}$.

Step 1: Canonical form

The line through $M_1(x_1,y_1,z_1)$ with direction $\{l,m,n\}$ is $\tfrac{x-x_1}{l}=\tfrac{y-y_1}{m}=\tfrac{z-z_1}{n}$: $\frac{x-1}{1}=\frac{y+1}{2}=\frac{z-1}{3}$.

Step 2: Parametric and vector form

\[ x=1+t,\quad y=-1+2t,\quad z=1+3t\qquad\big(\vec r=\vec r_1+t\vec p\big) \]

$t=0$ gives $M_1$; the direction is $\vec p$. For example, $t=1$ gives the point $(2,1,4)$.

Step 3: As two planes

Equating the fractions in pairs: $2x-y-3=0,\qquad 3x-z-2=0$.