Answer
It is proved that $ax+by+cz+d=0$ represents a equation of a plane and $n=\lt a,b,c\gt $ is a normal vector to the plane.
Work Step by Step
$ax+by+cz+d=0$
$(ax+d)+by+cz+d=0$
$a(x+d/a)+by+cz+d=0$
Re-write as the dot product.
$(ai+bj+zk)$ $\cdot$[$(x+d/a)i+(y-0)j+(z-0)k]=0$
$(ai+bj+zk)\cdot[(xi+yj+zk)-(d/ai+0j+0k)]=0$
This is an equation of a plane with normal vector $n=\lt a,b,c\gt $
Hence, the result is proved.
It is proved that $ax+by+cz+d=0$ represents a equation of a plane and $n=\lt a,b,c\gt $ is a normal vector to the plane.
