Write the plane through M_1(0,-1,0) containing…

Write the plane through $M_1(0,-1,0)$ containing $\tfrac{x-1}2=\tfrac y1=\tfrac{z+1}3$.

Explanation of Method

The plane through M₁(0, −1, 0) that contains a given line.

Take $A(1,0,-1)$ on the line: \[\vec n=\{2,1,3\}\times\overrightarrow{AM_1}=\{2,1,3\}\times\{-1,-1,1\}=\{4,-5,-1\}.\]

\[4x-5y-z-5=0\]

Check: $M_1$ gives $5-5=0$, $A$ gives $4+1-5=0$, and $\vec n\cdot\vec d=8-5-3=0$.

$M_1$ $A$

The plane $4x-5y-z-5=0$ contains the line (red) and the point $M_1$.

\[\boxed{\,4x-5y-z-5=0\,}\]