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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.

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