Explanation Given info: θ : Angle of indignation between unit tangent vector T ^ and the x − axis r → ( s ) Is are length parameterization θ = θ ( S ) Be the angle of inclination at r ( s ) Formula used: From the figure unit tangent vector can be written as T ^ = cos θ i ^ + sin θ j ^ = 〈 cos θ , sin θ 〉 θ is function of S So it can be rewrite as T ^ = 〈 cos θ ( s ) , sin θ ( s ) 〉 d T ∧ d s = d d s 〈 cos θ ( s ) , sin θ ( s ) 〉 Differentiation is distributive d T ∧ d s = 〈 d ( cos θ ( s ) ) d s , d ( sin θ ( s ) ) d s 〉 = 〈 d ( cos θ ( s ) ) d ( θ ( s ) ) × d θ ( s ) d s , d ( sin θ ( s ) ) d ( θ ( s ) ) × d ( Q ( s ) ) d s 〉 (Apply chain rule of differentiation)

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

See similar textbooks

Videos

Question

Chapter 14.4, Problem 71E

To determine

To prove:

Prove that $K (S) = | \frac{d θ}{d S} |$

Where $Q :$ angle indignation between unit tangent vectors $\hat{T} and the x - a x i s$

Assume that $\vec{r} (s)$ is are length parameterization and let $θ = θ (S)$ be the angle of indignation at $r (s)$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.4, Problem 70E

Chapter 14.4, Problem 72E

Students have asked these similar questions

Vector v = PQ has initial point P (2, 14) and terminal point Q (7, 3). Vector v = RS has initial point R (29, 8) and terminal point S (12, 17). Part A: Write u and v in linear form. Show all necessary work. Part B: Write u and v in trigonometric form. Show all necessary work. Part C: Find 7u − 4v. Show all necessary calculations.

An object is suspended by two cables attached at a single point. The force applied on one cable has a magnitude of 125 pounds and acts at an angle of 37°. The force on the other cable is 75 pounds at an angle of 150°.Part A: Write each vector in component form. Show all necessary work.Part B: Find the dot product of the vectors. Show all necessary calculations Part C: Use the dot product to find the angle between the cables. Round the answer to the nearest degree. Show all necessary calculations.

An airplane flies at 500 mph with a direction of 135° relative to the air. The plane experiences a wind that blows 60 mph with a direction of 60°.Part A: Write each of the vectors in linear form. Show all necessary calculations.Part B: Find the sum of the vectors. Show all necessary calculations. Part C: Find the true speed and direction of the airplane. Round the speed to the thousandths place and the direction to the nearest degree. Show all necessary calculations.

Chapter 14 Solutions

EBK CALCULUS

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To prove: Prove that K ( S ) = | d θ d S | Where Q : angle indignation between unit tangent vectors T ^ and the x − a x i s Assume that r → ( s ) is are length parameterization and let θ = θ ( S ) be the angle of indignation at r ( s )

Solution Summary: The author explains how to prove that K ( S ) = | d s | where Q : angle of indignation between unit tangent vectors T

Videos

To prove:

Prove that $K (S) = | \frac{d θ}{d S} |$

Where $Q :$ angle indignation between unit tangent vectors $\hat{T} and the x - a x i s$

Assume that $\vec{r} (s)$ is are length parameterization and let $θ = θ (S)$ be the angle of indignation at $r (s)$

Want to see the full answer?

Chapter 14 Solutions

To prove: Prove that K ( S ) = | d θ d S | Where Q : angle indignation between unit tangent vectors T ^ and the x − a x i s Assume that r → ( s ) is are length parameterization and let θ = θ ( S ) be the angle of indignation at r ( s )

Solution Summary: The author explains how to prove that K ( S ) = | d s | where Q : angle of indignation between unit tangent vectors T

Videos

To prove: Prove that K(S)=|dθdS| Where Q: angle indignation between unit tangent vectors T^ and the x−axis Assume that r→(s) is are length parameterization and let θ=θ(S) be the angle of indignation at r(s)

Want to see the full answer?

Chapter 14 Solutions

To prove:

Prove that $K (S) = | \frac{d θ}{d S} |$

Where $Q :$ angle indignation between unit tangent vectors $\hat{T} and the x - a x i s$

Assume that $\vec{r} (s)$ is are length parameterization and let $θ = θ (S)$ be the angle of indignation at $r (s)$