Vector question: dot product in cylindrical coordinates

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SUMMARY

The discussion focuses on calculating the dot product of vectors in cylindrical coordinates for the purpose of determining work done by a force. The user struggles with the application of the dot product formula in this coordinate system. A referenced resource suggests converting cylindrical coordinates to rectangular coordinates to simplify the calculation. The key takeaway is that the dot product in cylindrical coordinates requires careful consideration of vector components and angles, as the standard formula may not directly apply.

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Hello.

To find the work of a force, I have to perform a dot product between the force and a infinitesimal displacement. If they are in cylindrical coordinates, I can't manage to make the dot product.

Please, could you help me?

Thank you.
 
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But the author of this page seems to have used a conversion to rectangular coordinates to get the result.

I learned that dot product is the sum of the product of the components of the two vectors. But if you multiply the angles, what should it give? The sum of the product of the components does not hold for cylindrical coordinates?
 
The book claims the answer is that all the magnitudes are the same because "the gravitational force on the penguin is the same". I'm having trouble understanding this. I thought the buoyant force was equal to the weight of the fluid displaced. Weight depends on mass which depends on density. Therefore, due to the differing densities the buoyant force will be different in each case? Is this incorrect?

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