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.