Write the normal to the plane x-2y-z-2=0 through M_1(-2,1,1)

Write the normal to the plane $x-2y-z-2=0$ through $M_1(-2,1,1)$.

Explanation of Method

The normal to a plane through a given point, and its foot.

Step 1: Solution

\[\frac{x+2}{1}=\frac{y-1}{-2}=\frac{z-1}{-1}\]

Foot: substituting $x=-2+t,\ y=1-2t,\ z=1-t$ gives $6t-7=0$, so $t=\tfrac76$: $F\left(-\tfrac56,\ -\tfrac43,\ -\tfrac16\right),\qquad |M_1F|=\tfrac76\sqrt6$.

$M_1$ $F$

The normal through $M_1$ meets the plane $x-2y-z-2=0$ at $F\left(-\tfrac56,-\tfrac43,-\tfrac16\right)$.

Result

\[\boxed{\,\frac{x+2}{1}=\frac{y-1}{-2}=\frac{z-1}{-1}\,}\]

Remark

Correction: the source gives the foot as $\left(-\tfrac56,-\tfrac23,-\tfrac16\right)$. But $y=1-2\cdot\tfrac76=-\tfrac43$. Check: $-\tfrac56+\tfrac83+\tfrac16-2=0$. $\checkmark$