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Lets say you have a vector in spherical coordinates; how do you rewrite this vector into a cartesian one and vice versa?

Im fine with rewriting coordinates but vectors have got me confused. I've tried digging through info online but I couldn't find any good examples.

In the following task, Ive found the gradient of the Point P. I'm stuck at the last step trying to find the cartesian coordinates.

We should use the following equations to transform from spherical to cartesian coordinates but plugging in our data doesnt give the right answer? And also, since out inclination angle is zero shouldnt the resulting vector be pointing on only the Z-axis?

Im fine with rewriting coordinates but vectors have got me confused. I've tried digging through info online but I couldn't find any good examples.

In the following task, Ive found the gradient of the Point P. I'm stuck at the last step trying to find the cartesian coordinates.

We should use the following equations to transform from spherical to cartesian coordinates but plugging in our data doesnt give the right answer? And also, since out inclination angle is zero shouldnt the resulting vector be pointing on only the Z-axis?

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