Chapter 11: Q. 54 (page 890)

Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54–56.
$r (t) = <\cos t, \sin t, t>$

Short Answer

Expert verified

The torsion of the given vector is $τ (t) = \frac{1}{2} .$

Step by step solution

Step 1. Given Information.

The given vector function is $r (t) = <\cos t, \sin t, t> .$

Step 2. Compute the torsion.

To compute the torsion of the vector function, we will use the torsion formula $τ (t) = \frac{(r' (t) \times r'' (t)) \cdot r''' (t)}{{||r' (t) \times r'' (t)||}^{2}} .$

So,

$r (t) = <\cos t, \sin t, t> r' (t) = <- \sin t, \cos t, 1> r'' (t) = <- \cos t, - \sin t, 0> r''' (t) = <\sin t, - \cos t, 0> Now, r' (t) \times r'' (t) = |\begin{matrix} i & j & k \\ - \sin t & \cos t & 1 \\ - \cos t & - \sin t & 0 \end{matrix}| = i (\sin t) - j (\cos t) + k (1) = <\sin t, - \cos t, 1> Let' s compute the dot product ((r' (t) \times r'' (t)) \cdot r''' (t)), = <s i n t, - c o s t, 1> \cdot <s i n t, - c o s t, 0> = \sin^{2} t + \cos^{2} t = 1 . . . . . . . . (a) Now, ||r' (t) \times r'' (t)|| = \sqrt{{(sint)}^{2} + {(- cost)}^{2} + {(1)}^{2}} = \sqrt{1 + 1} = \sqrt{2} {||r' (t) \times r'' (t)||}^{2} = {(\sqrt{2})}^{2} {||r' (t) \times r'' (t)||}^{2} = 2 . . . . . . . . . . . (b)$

Step 3. Calculate.

Put the values of (a) and (b) in the torsion formula,

$τ (t) = \frac{(r' (t) \times r'' (t)) \cdot r''' (t)}{{||r' (t) \times r'' (t)||}^{2}} τ (t) = \frac{1}{2}$

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Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54–56.
$r (t) = <\cos t, \sin t, t>$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Compute the torsion.

Step 3. Calculate.

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Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54–56.r(t)=cost,sint,t

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Compute the torsion.

Step 3. Calculate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

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Study anywhere. Anytime. Across all devices.

Use the definition of torsion in Exercise 52 to compute the torsion of the vector functions in Exercises 54–56.
$r (t) = <\cos t, \sin t, t>$