   Chapter 13, Problem 11RE

Chapter
Section
Textbook Problem

# For the curve given by r(t) = ⟨sin3 t, cos3 t, sin2 t⟩, 0 ≤ t ≤ π/2, find(a) the unit tangent vector,(b) the unit normal vector,(c) the unit binomial vector, and(d) the curvature.

(a)

To determine

To find: The unit tangent vector T(t) for the curve r(t)=sin3t,cos3t,sin2t,0tπ2 .

Explanation

Formula used:

Write the expression to find the unit tangent vector T(t) for a curve with the vector function r(t) .

T(t)=r(t)|r(t)| (1)

Here,

r(t) is the tangent vector, which is the derivative of vector r(t) .

Write the required differential formulae to find the tangent vector r(t) as follows.

ddtsinnt=nsinn1tcostddtcosnt=ncosn1tsint

Calculation of tangent vector r(t) :

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=sin3t,cos3t,sin2t as follows.

ddt[r(t)]=ddtsin3t,cos3t,sin2tr(t)=ddt(sin3t),ddt(cos3t),ddt(sin2t)

Rewrite and compute the expression as follows.

r(t)=3sin31tcost,3cos31tsint,2sin21tcost=3sin2tcost,3cos2tsint,2sintcost

Calculation of unit tangent vector T(t) :

Substitute 3sin2tcost,3cos2tsint,2sintcost for r(t) in equation (1),

T(t)=3sin2tcost,3cos2tsint,2sintcost|3sin2tcost,3cos2tsint,2sintcost

(b)

To determine

To find: The unit normal vector N(t) for the curve r(t)=sin3t,cos3t,sin2t,0tπ2 .

(c)

To determine

To find: The unit binormal vector B(t) for a curve r(t)=sin3t,cos3t,sin2t,0tπ2 .

(d)

To determine

To find: The curvature k for a curve r(t)=sin3t,cos3t,sin2t,0tπ2 .

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### What are the two requirements for a random sample?

Statistics for The Behavioral Sciences (MindTap Course List)

#### Sketch the graph of the equation. 18. y = 2

Single Variable Calculus: Early Transcendentals

#### The vector represented by where A(4, 8) and B(6, 6)is:

Study Guide for Stewart's Multivariable Calculus, 8th

#### The graph at the right is the direction field for: a) y = x y b) y = xy c) y = x + y d) y = xy

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 