PROBLEM
The equation for the x-axis is y=0; slope: 0
The equation for the y-axis is x=0; slope: infinity
If the x-axis and y-axis are perpendicular, the product of the slopes must be –1, but in this case it is not true. How can we explain this? Are the x-axis and y-axis not perpendicular? SOLUTIONS
Seed Expert Claude Baudoin writes:
The product of zero times infinity is not defined, therefore it "can" be -1.
One way of looking at this is to consider two perpendicular lines, one very close to the x-axis, the other one very close to the y-axis, and to make them "tend" together toward these axes. For instance, the slope of the first one may be 0.01 and the slope of the other -100, so the product is -1. Keep rotating them closer to (respectively) horizontal and vertical. The slopes might now be 0.001 and -1000, and the product is still -1. As you keep rotating the two lines closer to the axes (the limits), the slopes become 0 and infinity. Since the product has been constant at -1, the limit of that product remains equal to -1.
Laurent van Roy adds:
In theory, zero multiplied by infinity is undetermined (as is zero divided by zero) so you cannot prove that the x-axis and y-axis are not perpendicular, but neither can you prove that they are not!
If a line is y=ax+b, then a perpendicular line will be y=(-1/a)x+c.
If you take the x-axis as the first line, its slope (a) is 0. The perpendicular one is y=(-1/0)x+c which has a slope of infinity.
I prefer to explain it this way:
If a line is ay = bx, then a perpendicular line will be by = -ax.
If you take the x-axis as the first line, you get b=0, and every perpendicular line is 0y=0=ax. Now you get the y-axis. This means that both are indeed perpendicular.
|
