# What is the product of the slopes of perpendicular lines and why is this product always negative?

kjcdb8er | Certified Educator

If the slope of a line is m, then the slope of a line perpendicular to it is -1/m.

So, m * -1/m = -1

This number is always negative because: a negative times a positive is always negative. If m is negative, then -1/m is positive.

neela | Student

Let the equation of the two lines be:

y=m1*x+c1 and

y=m2*x+c2, where m1 and m2 are the slopes of the lines and c1 and c2 are their  intercepts on the Y axis.

The angles A and B by the above two the lines with X axis is given by:

tanA = m1 and tan B = m2.

Therefore, the angle between the two lines, B-A is given by:

Tan(A-B) = (tanA-tanB)/{1+tanA*tanB}.............(1)

When A-B is a right angle  or 90 degrees, Tan(A-B) is tan 90 degrees , which should be infinite. Or the denominator on the right side of equation at (1)  is zero  or 1+tanA*tan B =0  or tanA*tan B=-1 or m1*m2=-1 or the product of the slope must be equal to minus one.

So,the  product of the slopes of the lines is mimus one, when they are at right angles or perpendicular to each other.