- #1
fishspawned
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Homework Statement
A defining property of a vector is that its components must transform in a particular fashio under a rotation. for a counterclockwise rotation around the z-axis, by and angle ∅ the components Ax, Ay, and Az of a vector A transform in the following fashion:
Ax --> Ax' = Axcos∅ + Aysin∅
Ay --> Ay' = -Axsin∅ + Aycos∅
Az --> Az' = Az
Show that the cross product A x B acts as a vector under a rotation about the z-axis
Homework Equations
see above
The Attempt at a Solution
i think what i am having trouble is knowing what i am aiming to show. so i have started out rather blindly, setting up some conditions and hoping that it will show me some clue.
i rotated the coordinate system for A and B until A sits directly along the positive X-Axis and B lies above it pointed up in the positive Z direction, diverging by an angle ∅. i was looking at A being rotated 90 degrees counterclockwise
Ax = A
Ay = 0
Az = 0
Bx = Bcos∅
By = 0
Bz = Bsin∅
this gave me a cross product of --> -ABsin∅ along the Y-axis
this didn't tell me anything
i am assuming that AxB is to be likened to a transformation of A but i am unsure how to proceed [am i supposed to be placing AxB straight up in the z-axis?