Orthogonal Coordinate System
Note that all these systems are right-handed:
Cylindrical Coordinate System:
Solution:
∴ the vector in cylindrical coordinates is:
Solution:
∴ 𝑃𝑃(−1.414, 1.685, 2)
Spherical Coordinate System
The vector in spherical coordinates is expressed as:
Relations between Cartesian & Spherical Coordinates
Transformation of vector from spherical coordinates to Cartesian coordinates
Transformation of vector from Cartesian coordinates to spherical coordinates
Solution:
𝑟𝑟 = �𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 = �12 + 32 + 22 = 3.742
𝑧𝑧
2
𝜃𝜃 = cos−1 = cos −1
= 57.692°
𝑟𝑟
3.742
𝜙𝜙 = tan−1
𝑦𝑦
3
= tan−1 = 71.565°
𝑥𝑥
1
∴ 𝑃𝑃(𝑟𝑟, 𝜃𝜃, 𝜙𝜙) = 𝑃𝑃(3.742, 57.692°, 71.565°)
Example 4: Find the vector A directed from (-1, 2, 1) to (-2, -3, -4) in
a) cartesian coordinates
b) cylindrical coordinates
c) spherical coordinates
Solution:
a) 𝐀𝐀 = (−2 − −1)𝐚𝐚𝐱𝐱 + (−3 − 2)𝐚𝐚𝐲𝐲 + (−4 − 1)𝐚𝐚𝐳𝐳
𝐀𝐀 = −𝐚𝐚𝐱𝐱 − 5𝐚𝐚𝐲𝐲 − 5𝐚𝐚𝐳𝐳
b) 𝐀𝐀 = 𝐴𝐴𝜌𝜌 𝐚𝐚𝛒𝛒 + 𝐴𝐴𝜙𝜙 𝐚𝐚𝛟𝛟 + 𝐴𝐴𝑧𝑧 𝐚𝐚𝐳𝐳
𝑥𝑥
𝜌𝜌
𝑦𝑦
𝜌𝜌
𝐴𝐴𝜌𝜌 = 𝐀𝐀 ∙ 𝐚𝐚𝛒𝛒 = 𝐴𝐴𝑥𝑥 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 + 𝐴𝐴𝑦𝑦 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑥𝑥 � � + 𝑦𝑦 � � ; 𝑏𝑏𝑏𝑏𝑏𝑏 𝜌𝜌 = �𝑥𝑥 2 + 𝑦𝑦 2
𝐴𝐴𝜌𝜌 = (−1) �
−1
�(−1)2 + (−5)2
� + (−5) �
−5
�(−1)2 + (−5)2
� = 5.1
𝐴𝐴𝜙𝜙 = 𝐀𝐀 ∙ 𝐚𝐚𝛟𝛟 = −𝐴𝐴𝑥𝑥 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 𝐴𝐴𝑦𝑦 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
−1
−5
𝐴𝐴𝜙𝜙 = −(−1) �
� + (−5) �
�=0
�(−1)2 + (−5)2
�(−1)2 + (−5)2
𝐴𝐴𝑧𝑧 = 𝐀𝐀 ∙ 𝐚𝐚𝐳𝐳 = 𝑧𝑧 = −5
∴ 𝐀𝐀 = 5.1𝐚𝐚𝛒𝛒 − 5𝐚𝐚𝐳𝐳
c) 𝐀𝐀 = 𝐴𝐴𝑟𝑟 𝐚𝐚𝐫𝐫 + 𝐴𝐴𝜃𝜃 𝐚𝐚𝛉𝛉 + 𝐴𝐴𝜙𝜙 𝐚𝐚𝛟𝛟
𝐴𝐴𝑟𝑟 = 𝐴𝐴𝑥𝑥 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃𝑐𝑐𝑐𝑐𝑐𝑐𝜙𝜙 + 𝐴𝐴𝑦𝑦 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠𝜙𝜙 + 𝐴𝐴𝑧𝑧 𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃
�(−1)2 + (−5)2
𝜌𝜌
�𝑥𝑥 2 + 𝑦𝑦 2
=
=
𝑟𝑟 �𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 �(−1)2 + (−5)2 + (−5)2
√26
= 714.006𝑚𝑚
𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 =
√51
𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃 =
𝑐𝑐𝑐𝑐𝑐𝑐𝜙𝜙 =
𝑠𝑠𝑠𝑠𝑠𝑠𝜙𝜙 =
𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃 =
𝑥𝑥
−1
=
= −196.116𝑚𝑚
𝜌𝜌 √26
𝑦𝑦
−5
=
= −980.581𝑚𝑚
𝜌𝜌 √26
𝑧𝑧
−5
=
= −700.140𝑚𝑚
𝑟𝑟 √51
Thus, 𝐴𝐴𝑟𝑟 = (−1)(714.006𝑚𝑚)(−196.116𝑚𝑚) + (−5)(714.006𝑚𝑚)(−980.581𝑚𝑚) + (−5)(−700.140𝑚𝑚)
𝐴𝐴𝑟𝑟 = 7.141
𝐴𝐴𝜃𝜃 = 𝐴𝐴𝑥𝑥 𝑐𝑐𝑜𝑜𝑜𝑜𝜃𝜃𝑐𝑐𝑐𝑐𝑐𝑐𝜙𝜙 + 𝐴𝐴𝑦𝑦 𝑐𝑐𝑜𝑜𝑜𝑜𝜃𝜃𝑠𝑠𝑠𝑠𝑠𝑠𝜙𝜙 − 𝐴𝐴𝑧𝑧 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃
𝐴𝐴𝜃𝜃 = (−1)(−700.140𝑚𝑚)( −196.116𝑚𝑚) + (−5)(−700.140𝑚𝑚)(−980.581𝑚𝑚) − (−5)(714.006𝑚𝑚)
𝐴𝐴𝜃𝜃 = 1.437𝜇𝜇
𝐴𝐴𝜙𝜙 = −𝐴𝐴𝑥𝑥 𝑠𝑠𝑠𝑠𝑠𝑠𝜙𝜙 + 𝐴𝐴𝑦𝑦 𝑐𝑐𝑐𝑐𝑐𝑐𝜙𝜙
𝐴𝐴𝜙𝜙 = (−1)(−980.581𝑚𝑚) + (−5)( −196.116𝑚𝑚) = 1.961
∴ 𝐀𝐀 = 7.141𝐚𝐚𝐫𝐫 + 1.437𝑥𝑥10−6 𝐚𝐚𝛉𝛉 + 1.961𝐚𝐚𝛟𝛟