Showing relationship between cartesian and spherical unit vectors

In summary, the conversation discusses the relationship between cartesian and spherical unit vectors. To find the spherical unit vectors in terms of the cartesian unit vectors, a set of three equations is needed, with one of them being \(\hat{e_r} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2+y^2+z^2}}\). The conversation ends with a question about finding equations for \(\hat{e_\phi}\) and \(\hat{e_\theta}\) and using them to solve for the cartesian unit vectors.
  • #1
skate_nerd
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I am asked to show that when \(\hat{e_r}\), \(\hat{e_\theta}\), and \(\hat{e_\phi}\) are unit vectors in spherical coordinates, that the cartesian unit vectors
$$\hat{i} = \sin{\phi}\cos{\theta}\hat{e_r} + \cos{\phi}\cos{\theta}\hat{e_\phi} - \sin{\theta}\hat{e_\theta}$$
$$\hat{j} = \sin{\phi}\sin{\theta}\hat{e_r} + \cos{\phi}\sin{\theta}\hat{e_\phi} - \cos{\theta}\hat{\theta}$$
$$\hat{k} = \cos{\phi}\hat{e_r} - \sin{\phi}\hat{e_\phi}$$
I'm having a bit of trouble figuring out where to start with this. I totally understand geometrically how to convert \(x\) \(y\) and \(z\) coordinates to spherical, but I feel like that isn't helping me here. How do I begin to find the spherical unit vectors in terms of the cartesian unit vectors, or vise versa?
 
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  • #2
Re: showing relationship between cartesian and spherical unit vectors

skatenerd said:
I am asked to show that when \(\hat{e_r}\), \(\hat{e_\theta}\), and \(\hat{e_\phi}\) are unit vectors in spherical coordinates, that the cartesian unit vectors
$$\hat{i} = \sin{\phi}\cos{\theta}\hat{e_r} + \cos{\phi}\cos{\theta}\hat{e_\phi} - \sin{\theta}\hat{e_\theta}$$
$$\hat{j} = \sin{\phi}\sin{\theta}\hat{e_r} + \cos{\phi}\sin{\theta}\hat{e_\phi} - \cos{\theta}\hat{\theta}$$
$$\hat{k} = \cos{\phi}\hat{e_r} - \sin{\phi}\hat{e_\phi}$$
I'm having a bit of trouble figuring out where to start with this. I totally understand geometrically how to convert \(x\) \(y\) and \(z\) coordinates to spherical, but I feel like that isn't helping me here. How do I begin to find the spherical unit vectors in terms of the cartesian unit vectors, or vise versa?

Well, let's see...

If we take a unit vector in the direction of r, we get:
$$\hat{e_r} = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{\sqrt{x^2+y^2+z^2}}$$
Yes?

Perhaps we can create a set of 3 equations like this?
Can you find an equation for $\hat{e_\phi}$ and $\hat{e_\theta}$?

After that we can try and solve it for the cartesian unit vectors!
 

FAQ: Showing relationship between cartesian and spherical unit vectors

How are cartesian and spherical unit vectors related?

Cartesian and spherical unit vectors are both types of coordinate systems used to describe the position and direction of objects in space. They are related through a mathematical transformation, where the cartesian coordinates (x, y, z) can be converted to spherical coordinates (r, θ, φ) using trigonometric functions.

What are the differences between cartesian and spherical unit vectors?

The main difference between cartesian and spherical unit vectors is their orientation and the way they measure direction. Cartesian unit vectors are aligned with the x, y, and z axes, while spherical unit vectors are aligned with the radial distance (r), zenith angle (θ), and azimuthal angle (φ). Additionally, cartesian unit vectors have a constant magnitude of 1, while the magnitude of spherical unit vectors varies depending on the direction.

Why are spherical unit vectors useful?

Spherical unit vectors are useful because they allow for a more natural and intuitive way of describing the position and direction of objects in 3D space. They are particularly useful in fields such as physics and engineering, where spherical coordinates are often used to describe the motion of objects in spherical systems, such as planets or satellites.

How can I convert between cartesian and spherical unit vectors?

To convert between cartesian and spherical unit vectors, you can use the following equations:

r = √(x² + y² + z²)

θ = arccos(z / r)

φ = arctan(y / x)

Where r is the radial distance, θ is the zenith angle, and φ is the azimuthal angle. Alternatively, you can use online calculators or specialized software to perform the conversion.

Can I use both cartesian and spherical unit vectors in the same problem?

Yes, it is possible to use both cartesian and spherical unit vectors in the same problem. In fact, it is common to use a combination of coordinate systems to describe the position and direction of objects in complex systems. However, it is important to ensure that the units and axes are consistent to avoid errors in calculations.

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