Chapter 11: Q. 14 (page 859)

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.

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

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

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

The graph is

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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.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding another parametrization.

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As we saw in Example 1, the graph of the vector-valued function r(t)=cost,sint,t,fort∈[0,2π] 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

Step by step solution

Step 1. Given Information.

Step 2. Finding another parametrization.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Probability and Statistics

Logic and Functions

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

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.