Prove: The unit normal vector n at point P is given by

Proof. The slope of the unit tangent vector t is

slope of t = m₁ = (dy/ds)/(dx/ds)

The unit normal vector n is perpendicular to t. If two lines are perpendicular their slopes are related by m₁m₂ = -1 where their slopes are m₁ and m₂. Thus if the slope of t is m₁, the slope of n must be given by

            slope of n = m₂ = -1/m₁ = - (dx/ds)/(dy/ds)

Consequently the expression for n is

since the real and imaginary parts of n define its slope.