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

ADVANCED METHODS

513

The lines are parallel if and only if 11, mi, nl are proportional
to 12, m2, n2.
If the lines are not parallel, there is a unique plane which
contains the first line and is parallel to or contains the second.
A normal to this plane is perpendicular to both lines and
hence has the direction components I mn2 |, I n112, I 11m2; cf.
Ch. XVIII, ~ 5. The equation of the plane is, then,
(1) | mln2 (X - x1)+ nll2 (y - y1) + 1 lm2 1 (z - z)= 0,
or
X - 1 Y -Y 2-1
(2) 1 l ml n = 0.
12 mn2 n2
If this plane contains the point P2 it will contain the entire
second line, and conversely. Consequently, the two non-parallel
lines intersect if and only if
X2-X1 Y2- Y1 z2- 1
(3) 11 mi nl = 0.
12 M2 n2
Equation of the Plane Determined by Two Intersecting or
Parallel Lines. If two given lines intersect or are parallel, the
plane in which they lie can be determined by one of the lines,
say the first, and a point of the second which does not lie on
the first. Its equation, then, can be found by the method of
~ 1. This method, though always applicable, is designed
primarily for the case when at least one of the lines is given
as the intersection of two planes.
If both lines are represented by continued equalities, the
plane which they determine has (2) as its equation, in case the
lines intersect. If the lines are parallel, a similar equation
for their plane can be found; cf. ~ 1, Ex. 15.
EXERCISES
Show that the given lines intersect or are parallel. In each
case find the equation of the plane which they determine.