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

THE STRAIGHT LINE

39

Since Y
0 = 02 - 1, /
it follows from Trigonometry that
t tan 02- tan 0 A
1 + tan 01 tan 02
and hence that / x
(1) tan = 1- X2. FIG. 6
1 + A1X2
The angle P is the angle from L1 to L2. That is, it is the
angle through which L1 must be rotated in the positive sense,
about the point A, in order that it coincide with L2. In particular, we agree to take it as the smallest such angle, always
less, then, than 180~: 0 < ( < 180~.*
If L1 and L2 are perpendicular, then, by (2), ~ 6, AX = - 1/X
and 1 + A1X2 = 0. Consequently, cot c, which is equal to the
reciprocal of the right-hand side of (1), has the value zero, and
so 5 = 90~.
Example. Let L 'and L2 be given by the equations,
L1: 4x - 2y +-7=0,
L2: 12x +4y- 5 = 0.
Here X1 = 2 and \2 =- 3, and (1) becomes
tan -3-2=1.
1 —6
L Vy Hence the angle 6 from L1 to L2 is 45~.
In deducing (1) it was assumed that L1 and
L2 both have slopes. If this is not the case,
at least one of the lines is parallel to the axis
of y and no formula is needed. The angle
L ( may be found directly. Suppose, for exFIG. 7 ample, that L1 and L2 are, respectively,
x+2=0 and x -y=.
* The figure shows L1 and L2 as intersecting lines, but formula (1) and
the deduction of it are valid also in case L1 and L2 are parallel. In this