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

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ANALYTIC GEOMETRY

form a directed trihedral (with edges of indefinite length)
which is right-handed by the test of Ch. XVII, ~ 5.
Ans. O, 0, 1.
8. The above problem, if the direction cosines of L1 and L2
are, respectively, 7 -,, - and, 3. Ans., -, 6.
6. Three Lines Parallel to a Plane. Given three lines, with
the direction components 11, mi, ni, 12, m2, n2, and 13, m3, n3.
The lines will be parallel to a plane or will lie in a plane,
if and only if there is a line, with the direction components
1, m, n, which is perpendicular to each of them, i.e. if and
only if
ll + mim + nln = 0,
(1) 121 + mgn + nm n = 0,
131 + m3m + n3n = 0.
But, by Ch. XVI, ~ 10, these three homogeneous linear equations have a solution for I, m, n, other than the obvious solution 0, 0, 0, when and only when the determinant of their
coefficients vanishes:
11 ml nl
(2) 12 m2 n2 =0
13 m3 n3
We have, therefore, the theorem: *
THEOREM. Three lines are parallel to a plane or lie in a
plane, if and only if the determinant of their direction components
has the value zero.
EXERCISES
1. Show that three lines through the origin-with the
direction components 2, - 1, 5, 3, 2, -4, 7, 0, 6 lie in a
plane.
* The proof covers not only the general case, when the given lines
have but one common perpendicular direction and the equations (1) a
one-parameter family of solutions, but also the special case in which the
given lines are parallel, when the lines have infinitely many common
perpendicular directions and the equations (1) a two-parameter family
of solutions.