Perpendicular Line

A line perpendicular to another is a line that intersects it at a right angle, that is, an angle of 90°. In the Cartesian plane, two lines are perpendicular when the product of their slope coefficients is equal to $ -1 $

Starting with a starting line in the form $ y = a x + b $, the slope of the perpendicular line is then $ c = -1/a $

All lines perpendicular to the first will therefore have an equation of the type: $ y = c x + d $, or $ y = (-1/a) x + d $

The parameter $ d $ (the y-intercept) can take any real value: it determines the vertical position of the line.

Thus, there are an infinite number of lines perpendicular to a given line, all parallel to each other.

If the first line has the equation $ y = ax + b $, and the second $ y = c x + d $, then they are perpendicular if and only if $ a \times c = -1 $

Identify the slope $ a $ of the starting line.

Calculate the slope $ c = -\frac{1}{a} $ of the perpendicular.

Knowing a point $ P(x_0, y_0) $ through which it passes, write the equation in reduced form: $ y - y_0 = c(x - x_0) $

Simplify to obtain the form $ y = cx + d $ with $ d = y_0 - cx_0 $

Solve the system formed by their equations: $ y = a x + b $ and $ y = c x + d $

By equating the two expressions, obtain the coordinates of the intersection point: $$ x = \frac{d-b}{a-c} $$ then calculate $ y $ by substituting into one of the equations.

If the starting line is vertical (equation of type $ x = c $), the perpendicular line is horizontal (equation $ y = k $) and the slope of a vertical line is undefined.

If the starting line is horizontal (equation of type $ y = c $), the perpendicular line is vertical (equation $ x = k $) and the slope of the horizontal line is zero.