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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 = m_{1} = (dy/ds)/(dx/ds)

The unit normal vector n is perpendicular to t. If two lines are perpendicular their slopes are
related by m_{1}m_{2} = -1 where their slopes are m_{1} and m_{2}. Thus if the slope of t is m_{1}, the slope of
n must be given by

slope of n = m_{2} = -1/m_{1} = - (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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