Thornton & Marion, Classical Dynamics, Fifth Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 08. An equation of a plane in vector form

show that the given equation is the equation of a plane.
This problem assumes
1) $\vec{A}$ be a vector from the origin to a fixed point $P$

2) $\vec{r}$ be a vector from the origin to a variabel point $Q(x_1,x_2,x_3)$
the equation of a plane, which is the form of

$a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$
Solution
Let the vector $\vec{A}$ be

$\vec{A} = (A_1, A_2, A_3)$
and the vector $\vec{r}$ be
$\vec{r} = (x_1, x_2, x_3)$
Then,
$\begin{array}{rcl} \vec{A} \cdot \vec{r} & = & A_1 x_1 + A_2 x_2 + A_3 x_3 \\ & = & A^2 \\ & = & A^2_1 + A^2_2 + A^2_3 \end{array}$
Thus,
$A_1 x_1 + A_2 x_2 + A_3 x_3 = A^2_1 + A^2_2 + A^2_3$
and it becomes
$A_1(x_1 - A_1) + A_2(x_2 - A_2) + A_3(x_3 - A_3) = 0$
It is the equation of a plane perpendicular to $\vec{A}$ and passing through the point $P$.