Math Problem Statement

Find the vectors T, N, and B at the given point.

r(t) = t2, 

2

3

 t3, t,   4, 

16

3

, 2 

Solution

It seems like the vector function r(t)\mathbf{r}(t) you provided is a bit unclear due to formatting issues. I'll interpret it based on standard notation for parametric curves. Let's assume the parametric vector function is:

r(t)=(t2,23t3,t)\mathbf{r}(t) = \left( t^2, \frac{2}{3}t^3, t \right)

This would be a curve in 3D space where ( \mathbf{r

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Math Problem Analysis

Mathematical Concepts

Vector Calculus
Tangent Vector
Normal Vector
Binormal Vector
Parametric Curves

Formulas

Tangent Vector: T(t) = r'(t) / |r'(t)|
Normal Vector: N(t) = T'(t) / |T'(t)|
Binormal Vector: B(t) = T(t) x N(t)

Theorems

Frenet-Serret Formulas

Suitable Grade Level

Undergraduate Level - Calculus III

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