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

kdgg0909gn 2021-11-16 Answered
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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Expert Answer

Feas1981
Answered 2021-11-17 Author has 1575 answers

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

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