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I need some advice on how to approach this problem -

I have one vector and its components given a particular frame of reference ($A_x,\, A_y,\, A_z$). I have a second vector's components that is computed in a different frame of reference ($B_x,\,B_y,\,B_z$) and I need the dot product between these two vectors. I can't simply take the dot product (i.e.,$\mathbf A\cdot\mathbf B=A_xB_x+A_yB_y+A_zB_z$) because the two vector's components are computed with different orientations in mind.

Given that there is no way to orient one vector to match the frame of reference of the other vector, is there a way I can still get a dot product?

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If you know how the two frames are related then yes: simply transform both vectors into the same basis. If you don't, then no.

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If you don't have any information that relates both frames of reference, then it is impossible to compute the dot product between both vectors. Imagine the following situation:

There is frame of reference A where $\boldsymbol{v_A}=\boldsymbol{\hat{i}_A}$, and frame of reference B where $\boldsymbol{v_B}=\boldsymbol{\hat{i}_B}$ .

If frame of reference B is let to have any relation with A, then $\boldsymbol{v_B}$ can be seen as anything in frame A. Different boosts will deform $\boldsymbol{v_B}$ in an infinite amount of ways as seen in frame A.

It would be analogous to finding a unique dot product between $\boldsymbol{v_A}$ and all other vectors in frame A.

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