Write the equation of the normal to the z -axis through…

Write the equation of the normal to the $z$-axis through $M_1(-2,-1,2)$.

Explanation of Method

The common normal from M₁(−2, −1, 2) to the z-axis.

Step 1: Solution

The foot on the $z$-axis is $F(0,0,2)$ (same height). The direction is $\overrightarrow{FM_1}=\{-2,-1,0\}\parallel\{2,1,0\}$, which is perpendicular to $\vec k$.

\[\frac x2=\frac y1,\quad z=2\qquad\Big(\text{equivalently } \tfrac{x+2}{-2}=\tfrac{y+1}{-1},\ z=2\Big)\]

The distance from $M_1$ to the axis is $|FM_1|=\sqrt5$.

$M_1(-2,-1,2)$ $F(0,0,2)$

The normal from $M_1$ to the $z$-axis lies in the plane $z=2$ and meets the axis at $F(0,0,2)$.

Result

\[\boxed{\,\frac x2=\frac y1,\quad z=2\qquad\Big(\text{equivalently } \tfrac{x+2}{-2}=\tfrac{y+1}{-1},\ z=2\Big)\,}\]