Hi.
I am asked to graph $\displaystyle \vec{r}(t)=2t\vec{i}+\sin(4t)\vec{j}+\cos(4t)\vec{ k}$. How do I go about this?
I'll assume that $\displaystyle \vec{i}$, $\displaystyle \vec{j}$ and $\displaystyle \vec{k}$ point along the x-, y- and z-axes, respectively. You see that along the path, $\displaystyle y^2+z^2=\sin^2 4t+\cos^2 4t=1$, so the graph lies on the cylinder with radius 1 going along the x-axis. The x-coordinate grows linearly. If x(t) were constant, then y(t), z(t) would follow a circle. Therefore, this is a spiral. Further, (y(t), z(t)) make 2 revolutions when t ranges from 0 to $\displaystyle \pi$, so when x(t) = 2t, there will be two revolutions when x ranges from 0 to $\displaystyle 2\pi$.
This is a spiral with the unit radius and the period $\displaystyle \frac{\pi}{2}$, whose center is the x-axis.
See this picture (red: $\displaystyle t=0\ldots\frac{\pi}{2}$, blue: $\displaystyle t=\frac{\pi}{2}\ldots\pi$)
For maple
with(plots):
A1 := spacecurve([2*t, sin(4*t), cos(4*t)], t=0..(1/2)*Pi, color=red, thickness=3, numpoints=1000):
A2 := spacecurve([2*t, sin(4*t), cos(4*t)], t=(1/2)*Pi..Pi, color=blue, thickness=3, numpoints=1000):
display(A1, A2, axes=normal);