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

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

Let r(t) be a differentiable vector valued function and $$\displaystyle{v}{\left({t}\right)}={r}{\left({t}\right)}$$ be the velocity vector. Then we define the unit tangent vector by as the unit vector in the direction of the velocity vector.
$$T(t)=\frac{v(t)}{|v(t)|}$$
Now, use the quotient rule to find T(t).
Then the principal unit normal vector N(t) is defined by
$$N(t)=\frac{v'(t)}{|v'(t)|}$$
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 curve
We proved that the principal unit normal vector N(t) point into the curve