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Prove that the dot product vw is unaffected by rotation between any pair of components.

Solution:

We consider a rotation between the x and y axes by angle s in the counterclockwise direction.
A vector having length L pointing in the direction of the x axis, will, after the rotation have component Lcos(x) in the x' direction and -Lsin(x) in the y' direction.
A vector with length M pointing in the direction of the y axis, will, after the rotation, have component Msin(s) in the x' direction and Mcos(s) in the y' direction.
We can use this information to determine the components in the x' and y' direction of any vector.
Suppose then that V has component v_x in the x direction and v_y in the y direction.
Then the rotation on V will have the same effect on v_x as on L and the same effect on v_y as on M.
We write this out explicitly both for V and a second arbitrary vector W.

v_{x '} = v_x cos(s) - v_y sin(s), v_{y '} = v_x sin(s) + v_y cos(s)
w_{x '} = w_x cos(s) - w_y sin(s), w_{y '} = w_x sin(s) + w_y cos(s).

The dot product before the rotation was v_x * w_x + v_y * w_y
After the rotation it becomes v_{x '} * w_x' + v_{y '} * w_y' which when written out consists of 8 terms.
There is v_x * w_x multiplied by cos(s) * cos(s) + sin(s) * sin(s), v_y * w_y multiplied by the same, and the two cross terms v_x * w_y and v_y * w_x, each multiplied by sin(s) * cos(s) - sin(s) * cos(s), that is multiplied by zero.
By the Pythagorian theorem, then, the dot product has not been changed by the rotation; that is to say, you can use the same procedure to evaluate it in either the original or rotated coordinates.