Thornton & Marion, Classical Dynamics of Particles and Systems, 5th 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. $latex \\vec{A}$ be a vector from the origin to a fixed point $latex P$
2. $latex \\vec{r}$ be a vector from the origin to a variabel point $latex Q(x_1,x_2,x_3)$

We should know about

• The equation of a plane, which is the form of

$latex a(x-x_0) + b(y-y_0) + c(z-z_0) = 0$

Solution

Let the vector $latex \\vec{A}$ be

$latex \\vec{A} = (A_1, A_2, A_3)$and the vector $latex \\vec{r}$ be

$latex \\vec{r} = (x_1, x_2, x_3)$ 

Then,

$latex \\begin{array}{rcl} \\vec{A} \\cdot \\vec{r} & = & A_1 x_1 + A_2 x_2 + A_3 x_3 \\\\ & = & A^2 \\\\ & = & A^2_1 + A^2_2 + A^2_3 \\end{array}$ 

Thus,

$latex A_1 x_1 + A_2 x_2 + A_3 x_3 = A^2_1 + A^2_2 + A^2_3$and it becomes

$latex A_1(x_1 – A_1) + A_2(x_2 – A_2) + A_3(x_3 – A_3) = 0$ 

It is the equation of a plane perpendicular to $latex \\vec{A}$ and passing through the point $latex P$.