Given a twice-differentiable vector-valued function r(t), why does the principal unit normal vector N(t) point into the curve? (Hint: Use the definition!)

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Feas1981 · Accepted Answer

Let r(t) be a differentiable vector valued function and&amp;nbsp;v(t)=r‘(t)&amp;nbsp;be the velocity vector. The unit vector in the direction of the velocity vector is then used to define the unit tangent vector.T(t)=v(t)|v(t)|Now, use the quotient rule to find T`(t).the definition of the principal unit normal vector N(t) follows:N(t)=v′(t)|v′(t)|Since the unit vector in the direction of a given vector will be the same after multiplying the vector by a positive scalar, we can simplify by multiplying by the factor.Geometrically, for a non straight curve, this vector is the unique vector that point into the curveWe proved that the principal unit normal vector N(t) point into the curve

Given a twice-differentiable vector-valued function r(t), why does the

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