It is a well known result from algebra that if two lines are perpendicular then the product of their slopes is always -1.^*

Give a very simple proof that if the lines form any angle besides 90^o the product of the slopes cannot be a constant.

^*Assuming neither line is vertical.

If the two lines are not perpendicular, then I can rotate them so that one line is horizontal and the other is neither horizontal nor vertical. The horizontal line has slope zero and so therefore is the product.

Now rotate the lines by some amount such that neither line is horizontal nor vertical. Clearly neither slope is zero now, and therefore neither is their product.

Since the product must be zero in one configuration and can't be zero in the other, it is not constant.

Perpendicular lines avoid this conflict because neither line can be made horizontal without the other being vertical, and so the zero slope case never "counts".

Posted by Paul on 2011-11-08 18:56:41