Why is the product of the slopes of two lines undefined if one of the lines is vertical?

Before we discuss this question it is worthwhile clarifying a common convention for specifying slope of a line.

1. The slope of a horizontal line in a direction of left to right is considered as not having any slope. Slopes of any other line is measured in terms of the angle that line makes with this horizontal lines.


2. The slope is measured in terms of either the measure of angle or tan of that angle. For this question we assume that the slope is being specified in term of tan of the angle of slope.

The slope of a vertical line, which mean a line making an angle of either 90 degrees or 270 degrees is +infinity in first case and -infinity in second case.

Also slope of a horizontal line is 0.

Any number other than 0 multiplied by infinity is equal to infinity. However, the product of 0 and infinity is undefined.

Therefore, when we multiply slopes of a vertical with a horizontal line the product is undefined.

For any line other than horizontal the product will be +infinity for the vertical line pointing upward and -infinity for vertical line pointing downwards.

It is worthwhile pointing out that if we consider products of slopes of two lines almost approaching vertical and horizontal instead of being perfect vertical and horizontal, the result will have value of either -1 or +1, depending on directions in which two lines point. When vertical line points up and horizontal line points right, or vertical line points down and horizontal line points left, the result will be +1. In other cases it will be -1.