Determine B and C in Ax+By+Cz+D=0 if the plane is parallel…

Determine $B$ and $C$ in $Ax+By+Cz+D=0$ if the plane is parallel to the $yOz$ plane.

Explanation of Method

A plane parallel to the yOz plane has zero coefficients B and C.

Step 1: Solution

Parallel planes have parallel normals. The plane $yOz$ ($x=0$) has normal $\vec i=\{1,0,0\}$, so $\{A,B,C\}\parallel\{1,0,0\}$: \[B=0,\quad C=0\qquad\Longrightarrow\qquad x=-\frac DA\quad(A\ne0)\]

The plane lies at distance $\left|\tfrac DA\right|$ from $yOz$. For $D=0$ it is $yOz$ itself.

$\vec n=\{A,0,0\}$ $x=-\tfrac DA$ $yOz$

A plane parallel to $yOz$ has normal parallel to $\vec i$, so $B=C=0$.

Result

\[\boxed{\,B=0,\quad C=0\qquad\Longrightarrow\qquad x=-\frac DA\quad(A\ne0)\,}\]