Sketch the curve represented by the vector-valued function and give the orientation of the curve.

Question

Sketch the curve represented by the vector-valued function  and give the orientation of the curve. ​

Quizplus · Accepted Answer

The given vector-valued function can be written in component form as  $ \langle t, \cos(3t), \sin(3t) \rangle$. The first component is simply linear, whereas the second and third components are functions of $ \cos(3t) $ and $ \sin(3t) $ respectively. Hence it indicates a parametrization of helix, which clearly narrows down our choices to C and D.

Now observe that the first two components of the given vector make a right-handed helix, i.e. one whose direction of rotation is counterclockwise, when viewed from the positive $ z $-axis. Therefore, the orientation of curve is counterclockwise. Among the remaining choices only C has a counterclockwise helix as the given vector-valued function. Hence the answer is (C).

Sketch the Curve Represented by the Vector-Valued Function and Give