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Does dividing by the magnitude change the direction of a vector?

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Multiplying or dividing by a scalar only changes the magnitude, not the direction.

One reason we use a unit tangent vector is because it is independent of the parameterization. A curve is a geometric object; a parameterization *of* that curve is a way to travel along the path, but whether you choose to go fast or slow is not an inherent geometric property of the path, and any one path can be parameterized in infinitely many ways.

Two different parameterizations of the same path can have different values of r' at a certain point - they will point in the same direction, but have different magnitudes, representing higher or lower speed. But they will have the same unit tangent vector, since that normalizes the speed away.

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