This is regarding 3d parametarization and vectors.

Question

Generally, I have a hard time conceptualizing how to sketch a vector that looks like $(\cos t, \sin t, t)$. How do I approach this? Usually, in an examination, there are really small bounds given so it's easy to just make a table of values, but is there a more intuitive way to do this?

Answer 1

I assume that the general pattern of your assignments is that you have a parameterized curve, i.e. one of your coordinates is always $t$. You can then interpret this coordinate as time and try to get an intuition what the remaining coordinates as function in $t$ are doing.

For your example $f:\ t\mapsto (\cos t,\sin t)$ describes the mapping of an angle to a point on the circle centered in (0,0) with radius 1. So if you start with $t=0$ and increase $t$ you "walk" along the circle counter-clockwise from beginning in (0,1).

To get the third dimension back in the picture imagine that you are also moving, let's say, upwards as you walk along the image of $f$. So that third coordinate in $(\cos t,\sin t,t)$ is now simultaneously interpreted as a direction as well as the time. To get a feel of that you can follow the curve with your pen and now also move your pen steadily upwards while you continue to follow the 2d curve. In your example what you should get is the outer railing of a spiral staircase or the edge of a corkscrew.