Write the equation of the plane that passes through the z…

Write the equation of the plane that passes through the $z$-axis and the point $M(2,1,1)$.

Step 1: Solution

A plane containing $Oz$ passes through $O$, so $D=0$, and its normal is perpendicular to $\vec k$, so $C=0$. Hence it has the form $Ax+By=0$. Through $M$: $2A+B=0$, so $B=-2A$.

\[x-2y=0\]

Check: every point $(0,0,t)$ satisfies it, and $M$ gives $2-2=0$. The normal $\{1,-2,0\}$ is perpendicular to $\vec k$. The $z$-coordinate of $M$ plays no role, since the plane is “vertical”.