Write the equation of the plane parallel to the z -axis that…

Write the equation of the plane parallel to the $z$-axis that cuts intercepts $2$ and $3$ on the $x$- and $y$-axes.

Explanation of Method

A plane parallel to the z-axis with intercept 2 on the x-axis and 3 on the y-axis.

Parallel to $Oz$ means $C=0$. With intercepts $2$ and $3$: $\frac x2+\frac y3=1$.

\[3x+2y-6=0\]

Check:

$y=z=0$ gives $x=2$, and $x=z=0$ gives $y=3$.
The normal $\{3,2,0\}$ is perpendicular to $\vec k$.
With every point $(x_0,y_0,z)$ the plane contains the whole vertical line through it.

$2$ $3$ $3x+2y=6$

The plane $3x+2y-6=0$ is parallel to $Oz$ and cuts intercepts $2$ and $3$ on the $x$- and $y$-axes.

\[\boxed{\,3x+2y-6=0\,}\]