Analytical Geometry

Hi, R.

This is true because the y axis has an undefined slope, so you can't
multiply it by anything. You can only talk about the product of the
slopes when both slopes exist, so your statement really needs that
qualification. Now, if we pretend infinity is a number, we find that
0*infinity is indeterminate, and -1 is one of the possible values it
can be considered to have, so that it does not contradict the theorem;
but it has no defined value. You can read about that concept in the
FAQ:

Dividing by Zero

But you raise an interesting point: is there a form of this fact that
applies to all pairs of lines, including those with infinite slope?

The best equation to use for a line that may have undefined slope is

ax + by = c

The slope, if it exists, is -a/b; when the slope is undefined, b = 0.

If we define two lines this way,

ax + by = c
dx + ey = f

then the product of the slopes will be -1 if

(-a/b)(-d/e) = -1

which can be rewritten as

ad = -be

This form does not require b and e to be non-zero, so it can be used
for all lines. The x and y axes are

0x + 1y = 0
1x + 0y = 0

so our condition is

0*1 = -1*0

which is indeed true. And if the first line is horizontal (a = 0), the
second line will only be perpendicular to it if e = 0, making it
vertical.