Question

As we saw in Example 1, the graph of the vector-valued function $r\left(t\right)=\left(\cos t,\sin t,t\right),\mathrm{for}t\in [0,2\pi ]$ is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion is downwards.

Short answer

Another parametrization for the given helix so that the motion is downwards is$r\left(t\right)=\left(\cos \left(-t\right),\sin \left(-t\right),\left(-t\right)\right),t\in \left[0,2\mathrm{\pi }\right].$

Step-by-step solution

Step 1. Given Information. 

The given vector-valued function is$r\left(t\right)=\left(\cos t,\sin t,t\right),\mathrm{for}t\in [0,2\pi ].$

Step 2. Finding another parametrization.

We have to find another parametrization for the given helix so that the motion is downwards.

For a circular helix that moves in a downward direction replace t by -t.

So, another parametrization is $r\left(t\right)=\left(\cos \left(-t\right),\sin \left(-t\right),\left(-t\right)\right),t\in \left[0,2\mathrm{\pi }\right].$

The graph is