# Transforming like a vector?

Problem: Find the transformation matrix between the coordinate systems C and C′ when C′ is obtained
i) by reflecting C in the plane x2 = 0
ii) by rotating C through a right angle about the axis OB, where B is the point with coordinates (2, 2, 1).

In each case, find the new coordinates of the point D whose coordinates in C are (3, -3, 0).

iii) Consider the force F = (1, -3, 2) in the system C, and the trans-
formation i) above. Show, by explicit calculation, that the moment (or torque) of the force about the point D above, OD × F , transforms like a vector.

Attempt at solution: I have no problem with any part of the problem except iii). I guess I just don't understand what it means to transform like a vector. I was thinking that I had to evaluate the cross product and then perform an arbitrary rotation about an arbitrary axis, and show that its magnitude is preserved, but that algebra gets incredibly messy. I think this is pretty simple, I just need to get a finger on what exactly I'm supposed to show.

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