Q. 56 Find all points of intersection ... [FREE SOLUTION]

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Chapter 11: Q. 56 (page 873)

Find all points of intersection between the graphs of the vector functions in Exercises 54鈥�56, and find the acute angle of intersection of the curves at those points.
$r_{1} (t) = <\cos t, \sin t, t>$ and $r_{2} (t) = <\sin t, \cos t, t>$

Short Answer

Expert verified

The point of intersection of $r_{1} (t)$ and $r_{2} (t)$ is $(\frac{\sqrt{2}}{2}, \frac{2}{2}, \frac{蟺}{4})$

The angle of intersection between the two curves is $90^{o}$

Step by step solution

Given Information

Consider $r_{1} (t) = <\cos t, \sin t, t>$ and $r_{2} (t) = <\sin t, \cos t, t>$

The objective is to find the points of intersection of $r_{1} (t)$ and $r_{2} (t)$

$r_{1} (t) = <\cos t, \sin t, t>$ and localid="1650567368707" $r_{2} (t) = <\sin t, \cos t, t>$

Since both $r_{1} (\frac{蟺}{4} = <\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, \frac{蟺}{4}>)$ and $r_{2} (\frac{蟺}{4}) = <\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, \frac{蟺}{4}>$

The two curves intersect at $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, \frac{蟺}{4})$

Thus , The point of intersection of $r_{1} (t)$ and $r_{1} (t)$ is $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, \frac{蟺}{4})$

Expression

The tangent vectors $t = \frac{蟺}{4}$

Let $u = <\frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 1>, v = <\frac{\sqrt{2}}{2}, \frac{- \sqrt{2}}{2}, 1>$

The cosine of the angle $胃$ between $r_{1} (t)$ and $r_{2} (t)$ is

$\cos 胃 = \frac{u . v}{||u|| ||v||} \cos 胃 = \frac{<\frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 1> <\frac{\sqrt{2}}{2}, \frac{- \sqrt{2}}{2}, 1>}{||<\frac{- \sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 1>|| ||<\frac{\sqrt{2}}{2}, \frac{- \sqrt{2}}{2}, 1>||} = \frac{\frac{- 1}{2} - \frac{1}{2} + 1}{\sqrt{\frac{1}{2} + \frac{1}{2} + 1} \sqrt{\frac{1}{2} + \frac{1}{2} + 1}} \cos 胃 = 0 胃 = \frac{蟺}{2}$

Thus at $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, \frac{蟺}{4})$ , the angle of intersection between the two curves is $90^{o}$

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91影视

Find all points of intersection between the graphs of the vector functions in Exercises 54鈥�56, and find the acute angle of intersection of the curves at those points.
$r_{1} (t) = <\cos t, \sin t, t>$ and $r_{2} (t) = <\sin t, \cos t, t>$

Short Answer

Step by step solution

Given Information

Expression

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Find all points of intersection between the graphs of the vector functions in Exercises 54鈥�56, and find the acute angle of intersection of the curves at those points.r1(t)=cost,sint,tand r2(t)=sint,cost,t

Short Answer

Step by step solution

Given Information

Expression

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Probability and Statistics

Statistics

Mechanics Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Find all points of intersection between the graphs of the vector functions in Exercises 54鈥�56, and find the acute angle of intersection of the curves at those points.
$r_{1} (t) = <\cos t, \sin t, t>$ and $r_{2} (t) = <\sin t, \cos t, t>$