Straight lines - product of slope of perpendicular lines.

Question

Let $l$ be a line with slope $\frac{1}{0}$, and $m$ a line with slope $0$. Now, line $l$ is perpendicular to line $l$, therefore the product of the slope of line $l$ and the slope of line $m$ equals $-1$. This implies that $\frac{1}{0} \times 0 = -1$!

Answer 1

As per wikipedia

"Two lines are parallel if and only if their slopes are equal and they are not the same line (coincident) or if they both are vertical and therefore both have undefined slopes. Two lines are perpendicular if the product of their slopes is −1 or one has a slope of 0 (a horizontal line) and the other has an undefined slope (a vertical line)."

So, it is obvious that the rule of multiplication of slopes is not applied when one of the lines is vertical and another horizontal. Further, y-axis (or more specifically, the line x = 0) doesn't have a slope of infinity but it is undefined.

Answer 2

In general, the equation for a line in the plane can be given in the following forms

• implicit

$$ax+by+c=0$$

which is more general, can describe all lines in the plane.

• explicit

$$y=-\frac a b x-\frac c b$$

which is less general from, we can't describe vertical lines in the plane, indeed the slope $m=-\frac a b$ is not well defined for $b=0$.