Write the equation of the plane through M_1(1,1,1) normal to…

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

Explanation of Method

The plane through M₁(1, 1, 1) perpendicular to the z-axis.

Step 1: Solution

The plane through $M_1(x_1,y_1,z_1)$ with normal $\vec n=\{A,B,C\}$ is $A(x-x_1)+B(y-y_1)+C(z-z_1)=0$. The normal is the direction of the $z$-axis, $\vec n=\vec k=\{0,0,1\}$:

\[ 0\cdot(x-1)+0\cdot(y-1)+1\cdot(z-1)=0 . \]

\[z=1\]

The plane is parallel to $xOy$ at height $1$, and meets the $z$-axis at $(0,0,1)$.

$\vec n=\vec k$ $M_1$ $1$ $z=1$

The plane $z=1$ through $M_1(1,1,1)$, perpendicular to the $z$-axis (normal $\vec k$).

Result

\[\boxed{\,z=1\,}\]