Find the piercing point of tfrac x2=tfracy-11=(z+1)/(-1)…

Find the piercing point of $\tfrac x2=\tfrac{y-1}1=\tfrac{z+1}{-1}$ through $x+y+3z-5=0$, and interpret geometrically.

Explanation of Method

A line that is parallel to a plane and never meets it.

Substituting $x=2t,\ y=1+t,\ z=-1-t$: $0\cdot t-7=0$, a contradiction. There is no piercing point.

Interpretation: $\vec n\cdot\vec d=2+1-3=0$, so the line is parallel to the plane. $M_0(0,1,-1)$ gives $-7\ne0$, so it does not lie in it.

\[\text{line}\parallel\text{plane},\qquad d=\frac{|-7|}{\sqrt{11}}=\frac{7}{\sqrt{11}}\]

$\tfrac{7}{\sqrt{11}}$ $M_0$

The line is parallel to the plane $x+y+3z-5=0$ at distance $\tfrac{7}{\sqrt{11}}$ (dashed): no piercing point.

\[\boxed{\,\text{line}\parallel\text{plane},\qquad d=\frac{|-7|}{\sqrt{11}}=\frac{7}{\sqrt{11}}\,}\]