Page 241 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Vector Form of Equation of a Plane

Vector notation provides a useful alternative way of writing the point-normal form of the equation of a plane: Referring to

Figure 3.5.3, let be the vector from the origin to the point , let be the vector from the

origin to the point , and let be a vector normal to the plane. Then , so Formula 1 can be

rewritten as

(5)

This is called the vector form of the equation of a plane.

Figure 3.5.3

EXAMPLE 3 Vector Equation of a Plane Using 5
The equation

is the vector equation of the plane that passes through the point and is perpendicular to the vector .

Lines in 3-Space

We shall now show how to obtain equations for lines in 3-space. Suppose that l is the line in 3-space through the point

and parallel to the nonzero vector . It is clear (Figure 3.5.4) that l consists precisely of those points

for which the vector is parallel to v—that is, for which there is a scalar t such that

(6)

In terms of components, (6) can be written as

from which it follows that , , and , so

As the parameter t varies from to , the point traces out the line l. The equations

(7)