Vector Calculus 4th Edition Chapter 1 Vectors
Page 73 Problem 1 Answer
Given: Points whose polar coordinates are (√3,5π/6)
To find: The Cartesian coordinates of the point.
Use the relation between polar and Cartesian coordinatesEvaluate to obtain the final answer
To convert from polar to Cartesian coordinates, use the following formula:
x = r cos θ
y = r sin θ
x = √3⋅cos 5 π/6
= √3⋅(−√3/2)
= −3/2
y = √3⋅sin 5 π/6
= √3⋅1/2
= √3/2
Hence the evaluated value is, (x,y)=(−3/2,√3/2)
Page 73 Problem 2 Answer
Given: Points whose polar coordinates is (3,0)
To find: The Cartesian coordinates of the point Use the relation between polar and Cartesian coordinates Evaluate to obtain the final answer
To convert from Polar to Cartesian coordinates, use the following formula:
x = r cosθ
y = r sinθ
So, x=3⋅cos0=3
y=3⋅sin0
=3⋅0
=0
Hence, the evaluated value is (x,y)=(3,0)
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Page 73 Problem 3 Answer
Given: Points whose Cartesian coordinates are (−2,2)
To find: The polar coordinates of the point Use the relation between polar and Cartesian coordinates Evaluate to obtain the final answer
To convert from Cartesian to Polar coordinates, use the following formula:
r2 = x2 + y2
tanθ = y/x
So, r2 = (−2)2 + 22
= 8
r = √8
= 2√2
tanθ = 2/−2
= −1
θ = 3π/4
Hence, the evaluated value is (r,θ)=(2√2,3π/4)
Page 73 Problem 4 Answer
Given: Points whose cylindrical coordinates are (2,2,2)
To find: The polar cartesian of the point the relation between cylindrical and Cartesian coordinatesEvaluate to obtain the final answer
The relation to converting from Cylindrical to Cartesian coordinates:
x = r cosθ
y = r sinθ
z = z
So, x = 2 cos2
y = 2 sin2
z = 2
Hence, the evaluated value is (x,y,z)=(2cos2,2sin2,2)
Page 73 Problem 5 Answer
Given: Points whose cylindrical coordinates are (π,π/2,1)
To find: The polar cartesian of the pointUse the relation between cylindrical and Cartesian coordinatesEvaluate to obtain the final answer
Use the following relation to converting from Cylindrical to Cartesian coordinates:
x = r cosθ
y = r sinθ
z = z
So,
x=π cos π/2=0
y = π sinπ/2
= π⋅1
= π
z = 1
Hence the evaluated value is, (x,y,z) = (0,π,1)
Page 73 Problem 6 Answer
Given: Cylindrical coordinates (1,2π/3,−2)
To find: The Cartesian coordinatesEvaluate to get the result.
Use following relation to convert from Cylindrical to Cartesian coordinates:
x = r cosθ
y = r sinθ
z = z
x = 1cos 2π/3
= 1⋅(−1/2)
=−1/2
y = 1sin 2 π/3
= 1⋅√3/2
= √3/2
z =−2
Therefore, these are the cartesian coordinates for the cylindrical coordinates:
(x,y,z)=(−1/2,√3/2,−2)
Page 73 Problem 7 Answer
Given: Spherical coordinates (4,π/2,π/3)
To find: The Rectangular coordinates Evaluate to get the result.
Use following relation to convert from spherical to rectangular (Cartesian) coordinates:
x = ρ sin φ cosθ
y = ρ sin φ sinθ
z = ρ cosφ
x = 4 sin π/2c osπ/3
= 4⋅1⋅1/2
= 2
y = 4 sin π/2 sinπ/3
= 4⋅1⋅√3/2
= 2√3
z=4cosπ/2
= 4⋅0
= 0
Therefore, these are the rectangular coordinates:
(x,y,z) = (2,2√3,0)
Page 73 Problem 8 Answer
Given: Spherical Coordinates (3,π/3,π/2)
To find: The Rectangular coordinates Evaluate to get the final result
Use following relation to convert from spherical to rectangular (Cartesian) coordinates:
x = ρ sin φ cosθ
y = ρ sin φ sinθ
z = ρ cos φ
x=3sinπ/3cosπ/2
=3⋅√3/2⋅0 =0
y = 3 si nπ/3 sinπ/2
= 3⋅√3/2⋅1
= 3√3/2
z = 3cosπ/3
=3⋅1/2
=3/2
Therefore,following are the rectangular coordinates of the above data:
(x,y,z) = (0,3√3/2,3/2)
Page 73 Problem 9 Answer
Given: Spherical coordinates (1,3π/4,2π/3)
To find: The Rectangular coordinates.
Evaluate to get the final answer
Use following relation to convert from spherical to rectangular (Cartesian) coordinates:
x= ρ sin φ cosθ
y = ρ sin φ sinθ
z = ρ cosφ
x = 1 sin3π/4cos2π/3
= √2/2⋅(−1/2)
= −√2/4
y = 1 sin 3π/4 sin2π/3
= √2/2⋅√3/2
= √6/4
z = 1cos3π/4
= −√2/2
Therefore,following are the rectangular coordinates of the above data:
(x,y,z) = (−√2/4,√6/4,−√2/2)
Page 73 Problem 10 Answer
Given: Spherical coordinates (2,π,π/4)
To find: The Rectangular coordinates.
Evaluate to get the final answer.
Use following relation to convert from spherical to rectangular (Cartesian) coordinates:
x = ρ sinφ cosθ
y =ρ sinφ sinθ
z =ρ cosφ
x = 2 sin π cosπ/4
= 2⋅0⋅√2/2
= 0
y = 2sinπsinπ/4
=2⋅0⋅√2/2
=0
z=2cosπ
=2⋅(−1)
=−2
Therefore,following are the rectangular coordinates of the above data:
(x,y,z)=(0,0,−2)
Page 73 Problem 11 Answer
Given : Cartesian Coordinates (−1,0,2)
To find: A set of cylindrical coordinates.
Evaluate to get the final result
Use the following relation to convert from Cartesian to cylindrical coordinates:
r2 = x2 + y2
tanθ = y/x
z=z
r2=(−1)2+02
= 1+0
= 1
r=1
tanθ = 0−1
=0
θ=π, because x coordinate is negative z=2
Therefore these are a set of cylindrical coordinates of the given data:
(r,θ,z)=(1,π,2)
Page 73 Problem 12 Answer
Given : Cartesian Coordinates (−1,√3,13)
To find: A set of cylindrical coordinates.
Evaluate to get the final result
Use following relation to convert from Cartesian to cylindrical coordinates:
r2 = x2 + y2
tanθ = y/x
z = z
r2 = (−1)2 + √3/2
=1+3
= 4
r = 2
tanθ = √3 −1
= − √3
θ= 2π/3, because x coordinate is negative
z =13
Therefore these are a set of cylindrical coordinates of the given data:
(r,θ,z) = (4,2π/3,13)