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Chapter 11: Q. 23 (page 889)
Find the arc length of the curves defined by the vector-valued functions on the specified intervals in Exercises 22鈥�27.
The length of the curve
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Evaluate the limits in Exercises 42鈥�45.
For constants , and , the graph of a vector-valued function of the form
As we saw in Example 1, the graph of the vector-valued function 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.
Every description of the DNA molecule says that the strands of the helices run in opposite directions. This is meant as a statement about chemistry, not about the geometric shape of the double helix. Consider two helices
(a) Sketch these two helices in the same coordinate system, and show that they run geometrically in different directions.
(b) Explain why it is impossible for these two helices to fail to intersect, and hence why they could not form a configuration for DNA.
Let and be differentiable vector functions with three components each. Prove that
(This is Theorem 11.11 (c).)What do you think about this solution?
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