- #1
yungman
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This is not a homework. Actual this is part of my own exercise on conversion where [itex] \vec A = \vec B \;X\; \vec C [/itex] and I intentionally set up B and C so the [itex] \theta_B \hbox { and } \theta_C \;=\; 60^o \;[/itex] respect to z-axis:
[tex] \vec B_{(x,y,z)} = (2,4, 2\sqrt{(\frac 5 3)}) \;\;\hbox { and }\;\; \vec C = (4,2, 2\sqrt{(\frac 5 3)}) [/tex]
[tex] \vec A_{(x,y,z)} = \hat x 4\sqrt{\frac 5 3} \;+\; \hat y 4\sqrt{\frac 5 3} -\hat z 12 [/tex]
I want to find A in spherical coordinate:
[tex] |\vec A_{(x,y,z)}| = \sqrt { x_A^2 +y_A^2 +z_A^2} = \sqrt{ 16\frac 5 3 \;+\; 16\frac 5 3 \;+\; 144} \;=\;14.04754 [/tex]
[tex] \phi = tan^{-1} (\frac y x) \;=\; 45^o \;\;\hbox { and }\;\; \theta = tan{-1}(\frac {\sqrt{x^2+y^2}}{z}) = tan{-1}(\frac {7.30297}{-12}) = -31.324^o \Rightarrow \theta = 180-31.324=148.676^o[/tex]
Or using cosine to find [itex] \theta [/itex]:
[tex]\theta = cos ^{-1}(\frac { z}{|\vec A|}) = cos ^{-1}( \frac {-12}{14.04754})= 148.676^o [/tex]
[tex]\Rightarrow \; cos \phi = sin\phi = 0.7071 \;\;\hbox { and }\;\; cos \theta = -0.85424 \;\;\hbox { and }\;\; sin \theta = 0.5198 [/tex]
Given equation of the [itex] \hat {\theta}[/itex] component:
[tex]A_{\theta} \;=\; A_x cos \theta cos \phi \;+\; A_y cos \theta sin\phi \;-\; A_z sin \theta [/tex]
[tex]\Rightarrow\; A_{\theta} \;=\; 4\sqrt{\frac 5 3} X (-0.85824) X 0.7071 \;+\; 4\sqrt{\frac 5 3} X (-0.85824) X 0.7071 \;-\;12 X (0.5198) \;=\; -6.23843 -6.2376 \;=\; -12.476 [/tex]
For a position vector:
[tex] \vec A \;=\; \hat R A_R [/tex]
Both [itex] A_{\theta } \;\;\hbox { and }\;\; A_{\phi} \;=\;0 [/itex], no exception! But as you can see, [itex] A_{\theta }[/itex] is not zero. I double check my work, I cannot see any error. Please tell me what am I missing.
Thanks
[tex] \vec B_{(x,y,z)} = (2,4, 2\sqrt{(\frac 5 3)}) \;\;\hbox { and }\;\; \vec C = (4,2, 2\sqrt{(\frac 5 3)}) [/tex]
[tex] \vec A_{(x,y,z)} = \hat x 4\sqrt{\frac 5 3} \;+\; \hat y 4\sqrt{\frac 5 3} -\hat z 12 [/tex]
I want to find A in spherical coordinate:
[tex] |\vec A_{(x,y,z)}| = \sqrt { x_A^2 +y_A^2 +z_A^2} = \sqrt{ 16\frac 5 3 \;+\; 16\frac 5 3 \;+\; 144} \;=\;14.04754 [/tex]
[tex] \phi = tan^{-1} (\frac y x) \;=\; 45^o \;\;\hbox { and }\;\; \theta = tan{-1}(\frac {\sqrt{x^2+y^2}}{z}) = tan{-1}(\frac {7.30297}{-12}) = -31.324^o \Rightarrow \theta = 180-31.324=148.676^o[/tex]
Or using cosine to find [itex] \theta [/itex]:
[tex]\theta = cos ^{-1}(\frac { z}{|\vec A|}) = cos ^{-1}( \frac {-12}{14.04754})= 148.676^o [/tex]
[tex]\Rightarrow \; cos \phi = sin\phi = 0.7071 \;\;\hbox { and }\;\; cos \theta = -0.85424 \;\;\hbox { and }\;\; sin \theta = 0.5198 [/tex]
Given equation of the [itex] \hat {\theta}[/itex] component:
[tex]A_{\theta} \;=\; A_x cos \theta cos \phi \;+\; A_y cos \theta sin\phi \;-\; A_z sin \theta [/tex]
[tex]\Rightarrow\; A_{\theta} \;=\; 4\sqrt{\frac 5 3} X (-0.85824) X 0.7071 \;+\; 4\sqrt{\frac 5 3} X (-0.85824) X 0.7071 \;-\;12 X (0.5198) \;=\; -6.23843 -6.2376 \;=\; -12.476 [/tex]
For a position vector:
[tex] \vec A \;=\; \hat R A_R [/tex]
Both [itex] A_{\theta } \;\;\hbox { and }\;\; A_{\phi} \;=\;0 [/itex], no exception! But as you can see, [itex] A_{\theta }[/itex] is not zero. I double check my work, I cannot see any error. Please tell me what am I missing.
Thanks
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