r(t) = 3t i + 5 sin(t) j + 5 cos(t) k

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.

Explore the vector function r(t) = 3t i + 5 sin(t) j + 5 cos(t) k, which describes particle motion in 3D space. Learn about position, velocity, and acceleration vectors through step-by-step differentiation, involving parametric equations and kinematic concepts. Suitable for Grades 11-12 or college-level calculus students.

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