Calculus: Analytic Geometry and Calculus, with Vectors

16 Analytic geometry in two dimensions

the slope-intercept formula. The easiest way to find the slope m of the

line having the equation 2x - 3y - 4 = 0 is to solve the equation for y

to obtain

y=Ix -I

and see that in = *.

Let L1 and Ls be two lines which are neither horizontal nor vertical and

let their slopes be ml and ms. Figure 1.34 reminds us of the elementary

fact in plane geometry that L1 and L2 are parallel if and only if 02 and 0,

are equal and hence if and only if tan 62 = tan 01 and ms = m1, Thus

L1 andL2 are parallel if and only if their slopes are equal.

L2 L1

Figure 1.34 Figure 1.35

For perpendicular lines, the story is more complicated. The lines L1

and L2 are perpendicular if and only if their slopes m1 and ms are negative

reciprocals, that is, ms = -1/m1 or m1 = -1/m2 or mlms = -1. To

prove this, we observe that L2 and L1 are perpendicular if and only if

02 = B1 + 7r/2 as in Figure 1.35 or 61 = 02 + 7r/2 when the roles of L1

and L2 are reversed. In the first case we have

(1.351) ms = tan (B1 -} 2) = -cot 01 = -

tan 01 ml

and the result follows. To get the result in the second case, we merely

T reverse the roles of L1 and L2.

Figure 1.36

x

As in t'igure 1.36, let L1 and L2 be two

nonvertical lines and let 0 (phi) be the

angle through which a line must be rotated

to bring it from coincidence with L1 to coin-

cidence (or parallelism) with L2. When we

can find the slopes m1 and ms of L1 and L2,

we can determine , from the fact that

0 = 7/2 when m1m2 = -1 and

(1.37) tan ¢ = ms - in,

1 + mlms

when mlms 96 -1. The latter formula is proved by the formula

(1.371) tan =tan (Bs -, B1) tan 02 - tan 01 ms - m1

= 1 -htan B1 tan Bs 1 -}- mlms