Explanation Proof: From the given graph, the direction of coordinate unit vectors in Cartesian coordinates remains same everywhere as the direction of standard coordinate unit vectors in polar coordinates changes with the change of ( r , θ ) . The unit vector along i ^ , it lies cos θ to u r and along the unit vector j ^ , it lies sin θ to u r . Thus, u r can be written as cos θ i ^ + sin θ j ^ (1) Similarly, along the unit vector − i ^ , their lies sin θ to u θ and along the unit vector j ^ , their lies cos θ to u θ . Thus, u θ can be written as − sin θ i ^ + cos θ j ^ (2) Hence, formed two powerful equations as u r = cos θ i ^ + sin θ j ^ and u θ = − sin θ i ^ + cos θ j ^ . Use elimination method to find the value of i ^ and j ^ . Multiply equations (1) and (2) by sin θ and cos θ , respectively. u r sin θ = sin θ cos θ i ^ + sin 2 θ j ^ u θ cos θ = − sin θ cos θ i ^ + cos 2 θ j ^ Add the above two equations . u r sin θ + u θ cos θ = sin θ cos θ i ^ − sin θ cos θ i ^ + sin 2 θ j ^ + cos 2 θ j ^ = ( sin θ cos θ − sin θ cos θ ) i ^ + ( sin 2 θ + cos 2 θ ) j ^ Here, j ^ = u r sin θ + u θ cos θ

2nd Edition

ISBN: 9781323178522

Author: Briggs

Publisher: PEARSON

See similar textbooks

Question

Chapter 14.1, Problem 56E

To determine

To show: The relationships

ur=cosθi^+sinθj^,i=urcosθ−uθ,sinθuθ=−sinθi^+cosθj^ and j=ursinθ+uθcosθ among the unit vectors in Cartesian and polar coordinate hold.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.1, Problem 55E

Chapter 14.1, Problem 57E

Students have asked these similar questions

The diagram below shows two vectors, i and v. 3 2 -2 -1 2 3 6 -2 -3 a. Consider vector ü from above. Write this vector in both component form and polar form. • Component form: o Polar form: Preview Preview b. Consider vector ö from above. Write this vector in both component form and polar form. • Component form: • Polar form: Preview Preview Hint: your answers should be in the form "". For the polar form, you do not need to enter a degree symbol. 5.

1. The cross winds hitting the plane due east is given by the formula as follows. 20 sin 2wt + 30 sin (3wt +) mph The driving wind (to a runway laying due North), directly behind the aircraft is given by the following formula. 10 cos wt - 30 cos (wt+ +7) mph If the course heading of the aircraft is due north, what is its resultant heading? Using vectors, plot the change required in course heading, and help the crew land the plane. You can ignore airspeed, turbulence etc... In addition, using software Model the combination of sine waves graphically and draw the graphical resultants of the waveforms and hence perform a graphical analysis as a comparison between your calculated results and what you have drawn.

Vector A is 3 units long at 25° and vector B is 3.5 units long at 105°. Draw the two vectors in a coordinate system schematically. Draw the sum of the two vectors in the same diagram. Calculate the x and y components of each vector. Calculate the x and y components of the vector C=A+B using the component method. .......................At time t=0 s, a stone is thrown from the top of a building horizontally with initial velocity 24 m/s. The building is so tall that the stone is still in the air at t=1.5 s. Calculate The horizontal displacement of the stone at 1=1.5 s. The vertical displacement of the stone at t=1.5 s

Chapter 14 Solutions

CODE/CALC ET 3-HOLE

Ch. 14.1 - How is a vector field F = f, g, h used to describe...Ch. 14.1 - Sketch the vector field F = x, y.Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Interpret the gradient field of the temperature...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...

Additional Engineering Textbook Solutions

Find more solutions based on key concepts

Identify f as being linear, quadratic, or neither. If f is quadratic, identify the leading coefficient a and ...

College Algebra with Modeling & Visualization (5th Edition)

In Exercises 13–16, find the margin of error for the values of c, ?, and n. 13. c = 0.95, ? = 5.2, n = 30

Elementary Statistics: Picturing the World (7th Edition)

Fill in each blank so that the resulting statement is true. An equation that expresses a relationship between t...

Algebra and Trigonometry (6th Edition)

The equivalent expression of x(y+z) by using the commutative property.

Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)

ASSESSMENT Each of the following sequences is either arithmetic or geometric. Identify the sequences 230,193+82...

A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)

In Problems 39-62, graph each function using the techniques of shifting, compressing, stretching, and/or reflec...

Precalculus

Knowledge Booster

The relationships u r = cos θ i ^ + sin θ j ^ , i = u r cos θ − u θ , sin θ u θ = − sin θ i ^ + cos θ j ^ and j = u r sin θ + u θ cos θ among the unit vectors in Cartesian and polar coordinate hold.

To show: The relationships ur=cosθi^+sinθj^,i=urcosθ−uθ,sinθuθ=−sinθi^+cosθj^ and j=ursinθ+uθcosθ among the unit vectors in Cartesian and polar coordinate hold.

Want to see the full answer?

Chapter 14 Solutions

Additional Engineering Textbook Solutions

To show: The relationships

ur=cosθi^+sinθj^,i=urcosθ−uθ,sinθuθ=−sinθi^+cosθj^ and j=ursinθ+uθcosθ among the unit vectors in Cartesian and polar coordinate hold.