How do you express unit vectors in spherical coordinates?

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Homework Help Overview

The discussion revolves around expressing Cartesian unit vectors (x hat, y hat, z hat) in terms of spherical unit vectors (r hat, theta hat, phi hat). The original poster is seeking clarification on the conversion process and has encountered difficulties in deriving the relationships between these vector representations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss visualizing the unit vectors and suggest using trigonometry to aid in understanding the relationships. There is mention of projecting vectors onto the xy plane and using polar coordinates for conversion. Some participants express uncertainty about specific expressions, such as whether r-hat cos theta is a valid representation.

Discussion Status

The discussion is ongoing, with various approaches being explored. Some participants have offered suggestions for visualizing the problem and using trigonometric relationships, while others express uncertainty about specific formulations. There is no explicit consensus yet on the correct expressions.

Contextual Notes

Participants are navigating the complexities of converting between coordinate systems and are working within the constraints of their current understanding of spherical coordinates and vector representations.

ak416
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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?
 
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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.
 
Is it just r-hat cos theta? I am not sure. Please tell me if anyone knows.
 

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