Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

504

ANALYTIC GEOMETRY

components 1, m, n can have its equation written in the form
X - X1 Y -Yi Z -Z1
x2- x1 Y2 -Y1 Z2- Z1 = 0.
I m n
Suggestion. Determine the direction components of the
normals to the plane.
2. Three Planes through a Line. Three Points on a Line.*
By means of the results of the preceding paragraph we can
prove the following theorem.
THEOREM 1. Three planes pass through a line or are parallel, when and only when any one of them is a linear combination of the other two.
If the three planes pass through a line, or are parallel, any
one of them passes through the line of intersection of the
other two, or is parallel to the other two. Consequently, by
Th. 3, ~ 1, this plane is a linear combination of the other two.
Conversely, if a particular one of the planes is a linear combination of the other two, it goes through the line of intersection of these two, if they intersect, or is parallel to them, if
they are parallel (Ths. 1, 2, ~ 1). It follows then, furthermore, by the first part of the proof, that any one of the three
planes is a linear combination of the other two, q. e. d.
For example, the three planes,
3x-2y+ z+ 6=0,
(1) 2x+5y-3z- 2=0,
4x- 9y+ Sz + 14=0,
pass through a line, inasmuch as the equation of the third can
be obtained by multiplying that of the first by 2, that of the
second by - 1, and by adding the results.
Three Points on a Line. The three points P1: (x1, yl, zl),
P2: (x2, Y2,z2), P3: (x3, y3, z3) lie on a line if and only if the
* It is assumed, here and in ~ 4, that the given planes, or the given
points, are distinct.