Page 239 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
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By multiplying out and collecting terms, we can rewrite 2 in the form
where a, b, c, and d are constants, and a, b, and c are not all zero. For example, the equation in Example 1 can be rewritten as
As the next theorem shows, planes in 3-space are represented by equations of the form .
THEOREM 3.5.1
If a, b, c, and d are constants and a, b, and c are not all zero, then the graph of the equation
(3)
is a plane having the vector as a normal.
Equation 3 is a linear equation in x, y, and z; it is called the general form of the equation of a plane.
Proof By hypothesis, the coefficients a, b, and c are not all zero. Assume, for the moment, that . Then the equation
can be rewritten in the form . But this is a point-normal form of the plane
passing through the point and having as a normal.
If , then either or . A straightforward modification of the above argument will handle these other cases.
Just as the solutions of a system of linear equations
correspond to points of intersection of the lines and in the -plane, so the solutions of a system
(4)
correspond to the points of intersection of the three planes , , and .
In Figure 3.5.2 we have illustrated the geometric possibilities that occur when 4 has zero, one, or infinitely many solutions.
