How do you express unit vectors in spherical coordinates?

[ak416]

Hi, I am having trouble with spherical coordinates. For example, how do you express the unit vectors x hat, y hat, z hat in terms of the spherical unit vectors r hat, theta hat, phi hat. I was able to go from spherical in terms of cartesian (with the help of mathworld.wolfram.com) but I can't get the other way. All i can think of is dR/dx / [dR/dx] = x hat where R is the position vectors in terms of spherical coords. I don't fully understand this but i saw this done as a derivation of theta hat in terms of x,y,z hat. So any suggestions?

 

[0rthodontist]

Do you visually know what they are? If you draw a picture you can use trigonometry. Phi is the angle down from the z axis, theta is the angle counterclockwise from the x axis, and r is the length of the vector. I'm sure you know how to change from polar coordinates to rectangular coordinates. Try this approach: project the vector straight down onto the xy plane so that you have theta and s, where s is the length of that projection, and don't worry about what s is exactly yet. You can find x and y in terms of theta and s by conversion from polar coordinates. Then if you can use trigonometry to find an expression for s in terms of r and phi, you can substitute that expression into your intermediate expressions for x and y. And you can find z easily straight from r and phi.

There are other ways of doing it, probably some of them are simpler.

 

[blueyellow]

Is it just r-hat cos theta? I am not sure. Please tell me if anyone knows.

 

[fluidistic]

Is it just r-hat cos theta? I am not sure. Please tell me if anyone knows.

Have a look at https://www.physicsforums.com/showthread.php?t=126702.