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

464

ANALYTIC GEOMETRY

planes would have no point in common. In either case the
hypothesis is contradicted. Hence we may state the theorem:
THEOREM. The three planes (1) intersect in a single point, if
and only if I A B C O 0.*
If [ A B C = -0, two or all three of the planes may be parallel;
these cases are easily detected by inspection of the equations.
Or, the three planes, taken in pairs, may intersect in three
distinct parallel lines. Or, finally, they may have a line in
common. We shall learn later, Ch. XXI, ~ 2, how to distinguish, from the equations of the planes, between these last
two cases.
EXERCISES
In each of the following exercises show that the three given
planes intersect in a single point, and find the coordinates of
the point.
1. The planes of Ch. XVI, ~ 2, Ex. 10.
2. The planes of Ch. XVI, ~ 2, Ex. 11.
3. The planes of Ch. XVI, ~ 2, Ex. 12.
4. The planes of Ch. XVI, ~ 2, Ex. 13.
5. Find the coordinates of the vertices of the tetrahedron
whose faces lie in the planes
z=0, 2y-3z=O, x-y+3=0, 5x-2y+3z=0.
Find the points of intersections of the following surfaces.
Draw a figure in each case.
6. x=4, z= - 2, x2 + y2=25.
7. x+y=2, x-y-=O, x2+z2-1=0.
8. x2+y2+z2=9, 5x+y-3z=5, x=z.
Ans. (2, 1, 2), (4, I, 4).
* Or, the three equations (1) have a unique solution, if and only if
A B C | = 0. This is the converse of Th. 10, Ch. XVI, ~ 8. We have
thus completed, by geometric methods, the proof of an important fact in
the theory of linear equations.