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.

As we are talking about slopes, it does ot matter if, for ease of discussion we translate the lines in question so their point of intersection is on the horizon, or, alternatively contruct a coordinate system with its origin at the point of intersection, which is basically the same thing.

Two lines that meet at other than a right angle will form two acute angles alternating with two obtuse angles. The acute ones will be easiest to talk about, and make the point.

When the rays of one the acute angles are each confined, by their particular phase of the rotation, to one qadrant of the translated origin, the product of their slopes will be positive.  When one is in one quadrant and the other is in another, the product of their slopes will be negative.  This is a smooth transition when one of the lines passes through the x-axis so the product is zero. As one of the lines approaches the y-axis, the product of the slopes gets larger in either the positive or negative direction without bound, as the other line is not approaching zero.

Posted by Charlie on 2011-11-08 13:03:17