How to approach graphing vector functions?

Question

Let's say I have the following vector function:

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

What properties of this function will allow me to sketch a curve drawn by this function?

I know that:

$x = t\cos t$

$y = t$

$z = t \sin t$

What is a general approach that I can take to solve problems that want me to sketch curves drawn by vector functions?

Answer 1

Trace out the space curve given by the parametrization $x=t\cos t$, $y=t$, $z=t\sin t$, for example, as $t$ varies over $[0,2\pi]$:

The vector-valued function $\mathbf{r}(t)=\langle t\cos t,t,t\sin t\rangle$ would be the function that at time $t$ outputs the vector $\langle t\cos t,t,t\sin t\rangle$, so at time $t$, its tip would be at the point $(t\cos t,t,t\sin t)$ on the space curve shown.