Thornton & Marion (5th Edition), Chapter 01, Exercise Problem 08

Thornton & Marion, Classical Dynamics, Fifth Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 08. An equation of a plane in vector form

The problem asks you to
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)
We should know about
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.

 

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