/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Find the unit tangent vector \(\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 15

Find the unit tangent vector \(\mathbf{T}\) and the curvature \(\kappa\) for the following parameterized curves. $$\mathbf{r}(t)=\langle\sqrt{3} \sin t, \sin t, 2 \cos t\rangle$$

Short Answer

Expert verified
Question: Find the unit tangent vector 饾悡 and curvature 饾渽 for the parameterized curve 饾惈(t) = 鉄ㄢ垰3 sin(t), sin(t), 2 cos(t)鉄�. Answer: To find the unit tangent vector 饾悡 and curvature 饾渽, we first compute the derivative of 饾惈(t) to get 饾惈'(t) and its magnitude. Then, we calculate 饾悡(t) by dividing 饾惈'(t) by its magnitude. Next, we differentiate 饾悡(t) and find the magnitude of 饾悡'(t). Finally, we find 饾渽(t) by dividing the magnitude of 饾悡'(t) by the magnitude of 饾惈'(t). Unit tangent vector 饾悡(t): 饾悡(t) = 鉄�(鈭�3 cos(t))/(鈭�(3 cos虏t + cos虏t + 4 sin虏t)), (cos(t))/(鈭�(3 cos虏t + cos虏t + 4 sin虏t)), (-2 sin(t))/(鈭�(3 cos虏t + cos虏t + 4 sin虏t))鉄� Curvature 饾渽(t): 饾渽(t) = (||饾悡'(t)||)/(||饾惈'(t)||) = (||鉄�(d/dt)((鈭�3 cos(t))/(鈭�(3 cos虏t + cos虏t + 4 sin虏t))), (d/dt)((cos(t))/(鈭�(3 cos虏t + cos虏t + 4 sin虏t))), (d/dt)((-2 sin(t))/(鈭�(3 cos虏t + cos虏t + 4 sin虏t)))鉄﹟|)/(鈭�(3 cos虏t + cos虏t + 4 sin虏t))

Step by step solution

01

1. Compute the derivative of \(\mathbf{r}(t)\)#

First, we need to find the tangent vector by differentiating the components of the parameterized curve with respect to \(t\): $$\mathbf{r'}(t) = \frac{d\mathbf{r}(t)}{dt} = \left\langle\frac{d(\sqrt{3}\sin t)}{dt}, \frac{d(\sin t)}{dt}, \frac{d(2\cos t)}{dt}\right\rangle = \langle\sqrt{3} \cos t, \cos t, -2 \sin t\rangle$$
02

2. Find the magnitude of \(\mathbf{r'}(t)\)#

Now, we find the magnitude of the tangent vector: $$||\mathbf{r'}(t)|| = \sqrt{(\sqrt{3} \cos t)^2 + (\cos t)^2 + (-2 \sin t)^2} = \sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}$$
03

3. Calculate the unit tangent vector \(\mathbf{T}(t)\)#

Now, we can find the unit tangent vector by dividing the tangent vector by its magnitude: $$\mathbf{T}(t) = \frac{\mathbf{r'}(t)}{||\mathbf{r'}(t)||} = \frac{\langle\sqrt{3} \cos t, \cos t, -2 \sin t\rangle}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}$$
04

4. Compute the derivative of the unit tangent vector \(\mathbf{T'}(t)\)#

Next, we differentiate the components of the unit tangent vector with respect to \(t\): $$\mathbf{T'}(t) = \frac{d\mathbf{T}(t)}{dt} = \left\langle\frac{d}{dt}\left(\frac{\sqrt{3} \cos t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right), \frac{d}{dt}\left(\frac{\cos t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right), \frac{d}{dt}\left(\frac{-2 \sin t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right)\right\rangle$$
05

5. Compute the magnitude of \(\mathbf{T'}(t)\)#

Now, we find the magnitude of the derivative of the unit tangent vector: $$||\mathbf{T'}(t)|| = \sqrt{\left(\frac{d}{dt}\left(\frac{\sqrt{3} \cos t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right)\right)^2 + \left(\frac{d}{dt}\left(\frac{\cos t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right)\right)^2 + \left(\frac{d}{dt}\left(\frac{-2 \sin t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right)\right)^2}$$
06

6. Calculate the curvature \(\kappa(t)\)#

Finally, we can find the curvature by dividing the magnitude of the derivative of the unit tangent vector by the magnitude of the tangent vector: $$\kappa(t) = \frac{||\mathbf{T'}(t)||}{||\mathbf{r'}(t)||} = \frac{\sqrt{\left(\frac{d}{dt}\left(\frac{\sqrt{3} \cos t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right)\right)^2 + \left(\frac{d}{dt}\left(\frac{\cos t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right)\right)^2 + \left(\frac{d}{dt}\left(\frac{-2 \sin t}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}\right)\right)^2}}{\sqrt{3 \cos^2 t + \cos^2 t + 4 \sin^2 t}}$$

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose water flows in a thin sheet over the \(x y\) -plane with a uniform velocity given by the vector \(\mathbf{v}=\langle 1,2\rangle ;\) this means that at all points of the plane, the velocity of the water has components \(1 \mathrm{m} / \mathrm{s}\) in the \(x\) -direction and \(2 \mathrm{m} / \mathrm{s}\) in the \(y\) -direction (see figure). Let \(C\) be an imaginary unit circle (that does not interfere with the flow). a. Show that at the point \((x, y)\) on the circle \(C\), the outwardpointing unit vector normal to \(C\) is \(\mathbf{n}=\langle x, y\rangle\) b. Show that at the point \((\cos \theta, \sin \theta)\) on the circle \(C,\) the outwardpointing unit vector normal to \(C\) is also \(\mathbf{n}=\langle\cos \theta, \sin \theta\rangle\) c. Find all points on \(C\) at which the velocity is normal to \(C\). d. Find all points on \(C\) at which the velocity is tangential to \(C\). e. At each point on \(C\), find the component of \(\mathbf{v}\) normal to \(C\). Express the answer as a function of \((x, y)\) and as a function of \(\theta\) f. What is the net flow through the circle? That is, does water accumulate inside the circle?

Consider the curve \(\mathbf{r}(t)=(a \cos t+b \sin t) \mathbf{i}+(c \cos t+d \sin t) \mathbf{j}+(e \cos t+f \sin t) \mathbf{k}\) where \(a, b, c, d, e,\) and \(f\) are real numbers. It can be shown that this curve lies in a plane. Graph the following curve and describe it. $$\begin{aligned}\mathbf{r}(t)=&(2 \cos t+2 \sin t) \mathbf{i}+(-\cos t+2 \sin t) \mathbf{j} \\\&+(\cos t-2 \sin t) \mathbf{k}\end{aligned}$$

An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t\rangle, \quad\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is a circle? b. What conditions on \(a, b, c,\) and \(d\) guarantee that the path is an ellipse?

Evaluate the following definite integrals. $$\int_{1 / 2}^{1}\left(\frac{3}{1+2 t} \mathbf{i}-\pi \csc ^{2}\left(\frac{\pi}{2} t\right) \mathbf{k}\right) d t$$

Jack pulls east on a rope attached to a camel with a force of 40 ib. Jill pulls north on a rope attached to the same camel with a force of 30 Ib. What is the magnitude and direction of the force on the camel? Assume the vectors lie in a horizontal plane.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.