To find the unit tangent vector T(t) for the parameterized curve r(t), we need to follow these steps:

Given Curve:

$\mathbf{r}(t) = \langle 8\cos(t), 8\sin(t), 7\cos(t) \rangle$ where $0 \leq t \leq \pi$.

Step 1: Find the derivative of r(t) (i.e., r'(t)):

To get the velocity vector (the derivative of the position vector):

[ \mathbf{r'}(t) = \left\langle \frac{d}{dt}[8\cos(t)], \frac{d}{dt}[8\sin(t)], \frac{d}{dt}[7\cos(t)] \right\rangle ] $\mathbf{r'}(t) = \langle -8\sin(t), 8\cos(t), -7\sin(t) \rangle$

Step 2: Find the magnitude of r'(t) (i.e., $|\mathbf{r'}(t)|$):

The magnitude of r'(t) is:

$|\mathbf{r'}(t)| = \sqrt{(-8\sin(t))^2 + (8\cos(t))^2 + (-7\sin(t))^2}$ $|\mathbf{r'}(t)| = \sqrt{64\sin^2(t) + 64\cos^2(t) + 49\sin^2(t)}$ $|\mathbf{r'}(t)| = \sqrt{64(\sin^2(t) + \cos^2(t)) + 49\sin^2(t)}$ Since $\sin^2(t) + \cos^2(t) = 1$, we simplify: $|\mathbf{r'}(t)| = \sqrt{64 + 49\sin^2(t)}$ Thus, the magnitude of the derivative is: $|\mathbf{r'}(t)| = \sqrt{64 + 49\sin^2(t)}$

Step 3: Find the unit tangent vector T(t):

The unit tangent vector is given by: $\mathbf{T}(t) = \frac{\mathbf{r'}(t)}{|\mathbf{r'}(t)|}$ Substituting the components: $\mathbf{T}(t) = \frac{\langle -8\sin(t), 8\cos(t), -7\sin(t) \rangle}{\sqrt{64 + 49\sin^2(t)}}$

So, the unit tangent vector is: $\mathbf{T}(t) = \left\langle \frac{-8\sin(t)}{\sqrt{64 + 49\sin^2(t)}}, \frac{8\cos(t)}{\sqrt{64 + 49\sin^2(t)}}, \frac{-7\sin(t)}{\sqrt{64 + 49\sin^2(t)}} \right\rangle$

Final Answer:

Thus, the correct unit tangent vector is:

$\mathbf{T}(t) = \left\langle \frac{-8\sin(t)}{\sqrt{64 + 49\sin^2(t)}}, \frac{8\cos(t)}{\sqrt{64 + 49\sin^2(t)}}, \frac{-7\sin(t)}{\sqrt{64 + 49\sin^2(t)}} \right\rangle$

Answer Choice:

A.

If you have any further questions or need more details, feel free to ask!

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Further Exploration:

1. How do we find the unit normal vector from the unit tangent vector?
2. How can we use the unit tangent vector to calculate the curvature of the curve?
3. What is the significance of the unit tangent vector in the context of motion along a curve?
4. Can the magnitude of r'(t) ever be zero? What would that imply?
5. How does the derivative of the curve change when we modify the parametric equations?

Tip:

When calculating the magnitude of r'(t), always simplify the expression using trigonometric identities like $\sin^2(t) + \cos^2(t) = 1$. This often makes the process easier.