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 ast increases. Find another parametrization for this helix so that the motion along the helix is faster for a given change in the parameter.

Short answer

Another parametrization for the given helix so that the motion along the helix is faster for a given change in the parameter is$r\left(t\right)=\left(\cos \left(21t\right),\sin \left(21t\right),t\right),\mathrm{for}t\in \left[0,2\mathrm{\pi }\right].$

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 this helix so that the motion along the helix is faster for a given change in the parameter.

Let another parametric curve is $r\left(t\right)=\left(\cos \left(21t\right),\sin \left(21t\right),t\right),\mathrm{for}t\in \left[0,2\mathrm{\pi }\right].$

So, the graph is

Thus, the motion along the helix is faster for a given change in the parameter.