Q. 25 Explain why the graph of every v... [FREE SOLUTION]

Chapter 11: Q. 25 (page 860)

Explain why the graph of every vector-valued function $r (t) = (\cos t, \sin t, \cos t)$ lies on the intersection of the two cylinders $x^{2} + y^{2} = 1 and y^{2} + z^{2} = 1 .$

Short Answer

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The graph of every vector-valued function $r (t) = (\cos t, \sin t, \cos t)$ lies on the intersection of the two cylinders $x^{2} + y^{2} = 1 and y^{2} + z^{2} = 1$ because r(t)satisfies both cylinders.

Step by step solution

Step 1. Given Information.

The given vector-valued function is $r (t) = (\cos t, \sin t, \cos t)$ and two cylinders are $x^{2} + y^{2} = 1 and y^{2} + z^{2} = 1 .$

Step 2. Explanation.

To explain why the graph of every vector-valued function lies pon the intersection of the given two cylinders, take the cylinder $x^{2} + y^{2} = 1 .$

Now, substitute $x = \cos t and y = \sin t$ in the above cylinder equation,

$x^{2} + y^{2} = 1 \cos^{2} t + \sin^{2} t = 1 1 = 1$

It is true.

Now, take the cylinder $y^{2} + z^{2} = 1 .$

Substitute $y = \sin t and z = \cos t$ in the above cylinder equation,

$y^{2} + z^{2} = 1 \sin^{2} t + \cos^{2} t = 1 1 = 1$

It is true.

Thus, r(t)satisfies both cylinders.

Hence the graph of every vector-valued function $r (t) = (\cos t, \sin t, \cos t)$ lies on the intersection of the two cylinders $x^{2} + y^{2} = 1 and y^{2} + z^{2} = 1 .$

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Explain why the graph of every vector-valued function $r (t) = (\cos t, \sin t, \cos t)$ lies on the intersection of the two cylinders $x^{2} + y^{2} = 1 and y^{2} + z^{2} = 1 .$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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Explain why the graph of every vector-valued function r(t)=cost,sint,costlies on the intersection of the two cylinders x2+y2=1andy2+z2=1.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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Most popular questions from this chapter

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Explain why the graph of every vector-valued function $r (t) = (\cos t, \sin t, \cos t)$ lies on the intersection of the two cylinders $x^{2} + y^{2} = 1 and y^{2} + z^{2} = 1 .$