The given vector function is:

$\mathbf{r}(t) = 3t\, \mathbf{i} + 5 \sin(t)\, \mathbf{j} + 5 \cos(t)\, \mathbf{k}$

This is a position vector in 3-dimensional space, where $t$ is a parameter and the components $i$, $j$, and $k$ represent the directions along the x, y, and z axes, respectively.

Components of $\mathbf{r}(t)$:

1. The $x$-component: $3t$
2. The $y$-component: $5 \sin(t)$
3. The $z$-component: $5 \cos(t)$

This vector describes the position of a particle as a function of time $t$. The term $3t$ describes uniform linear motion along the x-axis. The terms $5 \sin(t)$ and $5 \cos(t)$ describe circular motion in the yz-plane, with a radius of 5.

Interpretation:

• The particle moves linearly along the x-axis while simultaneously tracing out a circular path in the yz-plane.
• The circular motion comes from the trigonometric terms $\sin(t)$ and $\cos(t)$, which describe circular motion in a plane, with a constant radius of 5.

Derivatives:

1. Velocity vector $\mathbf{v}(t)$ is the derivative of $\mathbf{r}(t)$ with respect to $t$: $\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t) = 3\,\mathbf{i} + 5 \cos(t)\, \mathbf{j} - 5 \sin(t)\, \mathbf{k}$
2. Acceleration vector $\mathbf{a}(t)$ is the derivative of the velocity vector $\mathbf{v}(t)$ with respect to $t$: $\mathbf{a}(t) = \frac{d}{dt}\mathbf{v}(t) = -5 \sin(t)\, \mathbf{j} - 5 \cos(t)\, \mathbf{k}$

Do you want further details or any specific questions?
Here are 5 related questions:

1. What is the speed of the particle at time $t$?
2. How can the magnitude of the velocity vector be computed at a given time?
3. How does the particle's trajectory look in 3D space?
4. Can we find the position at a specific time $t = t_0$?
5. What happens to the particle's motion as $t \to \infty$?

Tip: The magnitude of a vector function is found by taking the square root of the sum of the squares of its components.