Q. 14 - Calculus | StudyQuestionHub

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Q. 14

Question

As we saw in Example 1, the graph of the vector-valued function $r (t) = (\cos t, \sin t, t), for t \in [0, 2 π]$ 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.

Step-by-Step Solution

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Answer

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

1Step 1. Given Information.

The given vector-valued function is $r (t) = (\cos t, \sin t, t), for t \in [0, 2 π] .$

2Step 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 $r (t) = (\cos (- t), \sin (- t), (- t)), t \in [0, 2 π] .$

The graph is

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