Determine the unit tangent vector to the curve whose parametric equations are: x=3cos t, y=3sin t, z=4t at the point (1, 2, 1).

Here are the solutions to the problems:

1. (a) Determine the unit tangent vector to the curve whose parametric equations are: $x=3\cos t, y=3\sin t, z=4t$ at the point $(1, 2, 1)$.

Step 1: Define the position vector $\mathbf{r}(t)$.

$$\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t \rangle$$

Step 2: Find the derivative of the position vector, $\mathbf{r}'(t)$. This is the tangent vector.

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