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 33
 Homework Statement:
 Find the basis vectors for the given coordinate system for the arbitrary constant ##a##.
 Relevant Equations:

The question is working with elliptic cylindrical coordinates. Refer to https://mathworld.wolfram.com/EllipticCylindricalCoordinates.html for more information.
Equations:
$$x = a \cosh{\mu} \cos{v}$$
$$y = a \sinh{\mu} \sin{v}$$
$$z=z$$
To my understanding, to get the basis vectors for a given coordinate system (in this case being the elliptic cylindrical coordinate system), I need to do something like shown below, right?
$$\hat{\mu}_x = \hat{\mu} \cdot \hat{x}$$
$$\hat{v}_z = \hat{v} \cdot \hat{z}$$
And do that for essentially 9 times in total. Despite that, though, how would I go about finding the actual resultant value? Also, perhaps this is more akin to a "semantic" question, what does it truly mean to find a basis vector for the arbitrary constant ##a##? Does it mean to find the basis vectors in terms of Cartesian basis vectors? Or perhaps something else entirely?
Ultimately, any help to assist me through the provided problem would be greatly appreciated. Thank you!
$$\hat{\mu}_x = \hat{\mu} \cdot \hat{x}$$
$$\hat{v}_z = \hat{v} \cdot \hat{z}$$
And do that for essentially 9 times in total. Despite that, though, how would I go about finding the actual resultant value? Also, perhaps this is more akin to a "semantic" question, what does it truly mean to find a basis vector for the arbitrary constant ##a##? Does it mean to find the basis vectors in terms of Cartesian basis vectors? Or perhaps something else entirely?
Ultimately, any help to assist me through the provided problem would be greatly appreciated. Thank you!