The given expression represents the position vector r(t) in three-dimensional space as a function of time t:

$\mathbf{r}(t) = \left( \cos(4t), \sin(4t), 4t \right)$

This vector function describes a curve in space where:

• The x-coordinate follows $\cos(4t)$,
• The y-coordinate follows $\sin(4t)$,
• The z-coordinate grows linearly with $4t$.

Key Insights:

1. The x and y components trace a circular motion in the xy-plane, with radius 1, as the cosine and sine functions oscillate between -1 and 1. The factor of 4 inside the cosine and sine functions means the motion completes one full cycle every $\frac{\pi}{2}$ units of time.
2. The z component increases linearly with time, making the motion spiral upwards as the curve progresses.

Would you like more details about the motion or how to analyze it further? Let me know if you have any specific questions.

Related Questions:

1. How does the speed of the particle change as it moves along this path?
2. How can we find the tangential velocity vector for this curve?
3. How does the curve look if we plot it in three dimensions?
4. What is the acceleration vector for this motion?
5. How would the motion change if the z-component were $\sin(4t)$ instead of $4t$?

Tip: The parametric equations like this one are useful for modeling circular and helical motions in physics, particularly when describing the movement of objects along curves.