Write the plane containing the parallel lines…

Write the plane containing the parallel lines $\tfrac{x-1}7=\tfrac{y-3}4=\tfrac{z-4}2$ and $\tfrac x7=\tfrac y4=\tfrac z2$.

Explanation of Method

The plane that contains two given parallel lines.

\[ \vec n=\{7,4,2\}\times\overrightarrow{OA}=\{7,4,2\}\times\{1,3,4\}=\{10,-26,17\}. \]

\[10x-26y+17z=0\]

Check: $A(1,3,4)$ gives $10-78+68=0$; $\vec n\cdot\vec d=70-104+34=0$.

$\ell_1$ $\ell_2$

Both parallel lines lie in the plane $10x-26y+17z=0$.

\[\boxed{\,10x-26y+17z=0\,}\]