Determine l in (x-1)/(l)=(y+2)/(2)=tfrac z-1 so that the…

Determine $l$ in $\tfrac{x-1}{l}=\tfrac{y+2}{2}=\tfrac z{-1}$ so that the line is parallel to $x-2y+3z-1=0$.

Explanation of Method

Choosing l so that the line is parallel to the plane.

Step 1: Solution

The condition is $\vec d\cdot\vec n=0$, i.e. $l-4-3=0$.

\[l=7\]

The base point $M_1(1,-2,0)$ gives $1+4-1=4\ne0$, so the line is strictly parallel, not lying in the plane. Its distance from the plane is $\tfrac{4}{\sqrt{14}}$.

$M_1$

For $l=7$ the line (red) is parallel to the plane $x-2y+3z-1=0$, at distance $\tfrac{4}{\sqrt{14}}$ (dashed).

Result

\[\boxed{\,l=7\,}\]