Find the unit tangent vector T(t) at the point with the given value of the parameter t.
r(t) = 4

t
 i + 2t2 j + 4t k,    t = 1

To find the unit tangent vector \( \mathbf{T}(t) \) at \( t = 1 \) for the vector function \( \mathbf{r}(t) = 4t \, \hat{i} + 2t^2 \, \hat{j} + 4t \, \hat{k} \), follow these steps:

### 1. Compute the derivative \( \mathbf{r}'(t) \) (velocity vector).
Take the derivative of each component of \( \mathbf{r}(t) \):

\[
\mathbf{r}'(t) = \frac{d}{dt}(4t \, \hat{i} + 2t^2 \, \hat{j} + 4t \, \hat{k})
\]

\[
\mathbf{r}'(t) = 4 \, \hat{i} + 4t \, \hat{j} + 4 \, \hat{k}
\]

### 2. Evaluate \( \mathbf{r}'(t) \) at \( t = 1 \):

\[
\mathbf{r}'(1) = 4 \, \hat{i} + 4(1) \, \hat{j} + 4 \, \hat{k}
\]
\[
\mathbf{r}'(1) = 4 \, \hat{i} + 4 \, \hat{j} + 4 \, \hat{k}
\]
So, the velocity vector at \( t = 1 \) is:
\[
\mathbf{r}'(1) = \langle 4, 4, 4 \rangle
\]

### 3. Compute the magnitude of \( \mathbf{r}'(1) \):

\[
|\mathbf{r}'(1)| = \sqrt{4^2 + 4^2 + 4^2} = \sqrt{16 + 16 + 16} = \sqrt{48} = 4\sqrt{3}
\]

### 4. Compute the unit tangent vector \( \mathbf{T}(t) \):
The unit tangent vector is given by:

\[
\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}
\]

At \( t = 1 \), this becomes:

\[
\mathbf{T}(1) = \frac{\langle 4, 4, 4 \rangle}{4\sqrt{3}} = \left\langle \frac{4}{4\sqrt{3}}, \frac{4}{4\sqrt{3}}, \frac{4}{4\sqrt{3}} \right\rangle = \left\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right\rangle
\]

Thus, the unit tangent vector at \( t = 1 \) is:

\[
\mathbf{T}(1) = \left\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right\rangle
\]

Would you like more details or further explanations?

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Here are 5 related questions for further exploration:

1. How do you find the curvature \( \kappa(t) \) of a vector function \( \mathbf{r}(t) \)?
2. What is the relationship between the unit tangent vector \( \mathbf{T}(t) \) and the unit normal vector \( \mathbf{N}(t) \)?
3. How can you compute the arc length of a curve given by \( \mathbf{r}(t) \)?
4. What is the significance of the binormal vector \( \mathbf{B}(t) \) in the Frenet-Serret frame?
5. How do you find the acceleration vector from the position vector \( \mathbf{r}(t) \)?

**Tip**: The unit tangent vector is useful for understanding the direction of motion along a curve, and it is always tangent to the curve at the point of interest.

Find the unit tangent vector T(t) at the point with the given value of the parameter t. r(t) = 4t i + 2t^2 j + 4t k, t = 1

Learn how to find the unit tangent vector for the vector function r(t) = 4t i + 2t^2 j + 4t k at t = 1. This solution includes detailed steps such as calculating the derivative, magnitude, and unit tangent vector. Suitable for high school and college students studying vector calculus.

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