What is the coefficient D in Ax+By+Cz+D=0 if the plane…

What is the coefficient $D$ in $Ax+By+Cz+D=0$ if the plane passes through the origin?

Step 1: Solution

Substitute $O(0,0,0)$: $A\cdot0+B\cdot0+C\cdot0+D=0$.

\[D=0:\qquad Ax+By+Cz=0,\quad (A,B,C)\ne(0,0,0)\]

Conversely, if $D=0$ then $O$ satisfies the equation. So $D=0$ is necessary and sufficient. In vector form such a plane is $\vec n\cdot\vec r=0$: all position vectors perpendicular to $\vec n$.