Chapter 11: Q. 25 (page 880)

For each of the vector-valued functions in Exercises 22–28, find the unit tangent vector.
$r (t) = (\cos^{3} t, \sin^{3} t)$

Short Answer

Expert verified

$\frac{1}{\sin t \cos t} <- \cos^{2} t \sin t, \sin^{2} t \cos t>$

Step by step solution

Step1. Given Information

$Consider r (t) = <\cos^{3} t, \sin^{3} t> The objective is to find the unit tangent vector to r (t) The unit tangent vector The unit tangent vector of r (t) denoted by T (t) is defined as T (t) = \frac{r^{'} (t)}{∥r^{'} (t)∥} Consider r (t) = <\cos^{3} t, \sin^{3} t> First we compute r^{'} (t) : \begin{array}{l} r^{'} (t) = \frac{d}{d t} <\cos^{3} t, \sin^{3} t> \\ = <\frac{d}{d t} \cos^{3} t, \frac{d}{d t} \sin^{3} t> \\ = <- 3 \cos^{2} t \sin t, 3 \sin^{2} t \cos t> \\ = 3 <- \cos^{2} t \sin t, \sin^{2} t \cos t> \end{array}$

Step2. Continue

$\begin{array}{l} ∥r^{'} (t)∥ = ∥<- 3 \cos^{2} t \sin t, 3 \sin^{2} t \cos t>∥ \\ = \sqrt{{(- 3 \cos^{2} t \sin t)}^{2} + {(3 \sin^{2} t \cos t)}^{2}} \\ = 3 \sqrt{\cos^{4} t \sin^{2} t + \sin^{4} t \cos^{2} t} \\ = 3 \sqrt{\sin^{2} t \cos^{2} t (\cos^{2} t + \sin^{2} t)} \\ = 3 \sqrt{\sin^{2} t \cos^{2} t} \sin c e \sin^{2} t + \cos^{2} t = 1 \\ = 3 \sin t \cos t \\ The unit tangent vector to r (t) is \\ \begin{array}{l} T (t) = \frac{r^{'} (t)}{∥r^{''} (t)∥} \\ = \frac{3 <- \cos^{2} t \sin t, \sin^{2} t \cos t>}{3 \sin t \cos t} \\ = \frac{1}{\sin t \cos t} <- \cos^{2} t \sin t, \sin^{2} t \cos t> \\ Thus the unit tangent vector to the given vector function is \\ \frac{1}{\sin t \cos t} <- \cos^{2} t \sin t, \sin^{2} t \cos t> \end{array} \end{array}$

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For each of the vector-valued functions in Exercises 22–28, find the unit tangent vector.
$r (t) = (\cos^{3} t, \sin^{3} t)$

Short Answer

Step by step solution

Step1. Given Information

Step2. Continue

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For each of the vector-valued functions in Exercises 22–28, find the unit tangent vector.r(t)=(cos3t,sin3t)

Short Answer

Step by step solution

Step1. Given Information

Step2. Continue

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Study anywhere. Anytime. Across all devices.

For each of the vector-valued functions in Exercises 22–28, find the unit tangent vector.
$r (t) = (\cos^{3} t, \sin^{3} t)$