Explanation Check the equation u r = cos θ i ^ + sin θ j ^ is consistent when θ = 0 , π 2 , π and 3 π 2 . When θ = 0 , u r = cos 0 i ^ + sin 0 j ^ = 1 ⋅ i ^ + 0 ⋅ j ^ = i ^ When θ = π 2 , u r = cos π 2 i ^ + sin π 2 j ^ = 0 ⋅ i ^ + 1 ⋅ j ^ = j ^ When θ = π , u r = cos π i ^ + sin π j ^ = − 1 ⋅ i ^ + 0 ⋅ j ^ = − i ^ When θ = 3 π 2 , u r = cos 3 π 2 i ^ + sin 3 π 2 j ^ = 0 ⋅ i ^ + − 1 ⋅ j ^ = − j ^ Check for the equation u θ = − sin θ i ^ + cos θ j ^ is consistent when θ = 0 , π 2 , π and 3 π 2 . When θ = 0 , u θ = − sin 0 i ^ + cos 0 j ^ = 0 ⋅ i ^ + 1 ⋅ j ^ = j ^ When θ = π 2 , u θ = − sin π 2 i ^ + cos π 2 j ^ = − 1 ⋅ i ^ + 0 ⋅ j ^ = − i ^ . When θ = π , u θ = − sin π i ^ + cos π j ^ = 0 ⋅ i ^ + ( − 1 ) ⋅ j ^ = − j ^ . When θ = 3 π 2 , u θ = − sin 3 π 2 i ^ + cos 3 π 2 j ^ = 1 ⋅ i ^ + 0 ⋅ j ^ = i ^ . Check for the equation and i = u r cos θ − u θ sin θ is consistent when θ = 0 , π 2 , π and 3 π 2

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

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Question

Chapter 14.1, Problem 57E

To determine

To show: The following relationships among the unit vectors in Cartesian and polar coordinates:

ur=cosθi^+sinθj^, i=urcosθ−uθsinθ, uθ=−sinθi^+cosθj^ and j=ursinθ+uθcosθ

are consistent with θ=0,π2,π and 3π2 .

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Chapter 14.1, Problem 56E

Chapter 14.1, Problem 58E

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Chapter 14 Solutions

Calculus: Early Transcendentals (2nd Edition)

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Knowledge Booster

The following relationships among the unit vectors in Cartesian and polar coordinates: 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 are consistent with θ = 0 , π 2 , π and 3 π 2 .

Solution Summary: The author explains the following relationships among the unit vectors in Cartesian and polar coordinates: u_r=mathrmcostheta

To show: The following relationships among the unit vectors in Cartesian and polar coordinates:

ur=cosθi^+sinθj^, i=urcosθ−uθsinθ, uθ=−sinθi^+cosθj^ and j=ursinθ+uθcosθ

are consistent with θ=0,π2,π and 3π2 .

Want to see the full answer?

Chapter 14 Solutions

The following relationships among the unit vectors in Cartesian and polar coordinates: 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 are consistent with θ = 0 , π 2 , π and 3 π 2 .

Solution Summary: The author explains the following relationships among the unit vectors in Cartesian and polar coordinates: u_r=mathrmcostheta

To show: The following relationships among the unit vectors in Cartesian and polar coordinates: ur=cosθi^+sinθj^, i=urcosθ−uθsinθ, uθ=−sinθi^+cosθj^ and j=ursinθ+uθcosθ are consistent with θ=0,π2,π and 3π2 .

Want to see the full answer?

Chapter 14 Solutions

To show: The following relationships among the unit vectors in Cartesian and polar coordinates:

ur=cosθi^+sinθj^, i=urcosθ−uθsinθ, uθ=−sinθi^+cosθj^ and j=ursinθ+uθcosθ

are consistent with θ=0,π2,π and 3π2 .