# 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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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.

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'm pretty sure you're not meant to do that, it just isn't consistent with the rest of the question, and in particular with the "explicit calcualation" instruction

but i've no idea what they do want

to make things worse, a moment (or any cross product ) isn't a vector anyway … it's a pseudovector, which "goes the wrong way" compared with vectors when you reflect it (as in operation (i))!