Rotation around the z-axis and cross products

[fishspawned]

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?

 

[tiny-tim]

hi fishspawned!

… i think what i am having trouble is knowing what i am aiming to show …

A is a vector (A_x,A_y,A_z)

if you rotate it, it becomes a vector A' = (A'_x,A'_y,A'_z)

where A'_x = = A_xcos∅ + A_ysin∅ etc

you have to prove that if A x B = C, then A' x B' = C' ​

 

[fishspawned]

thanks so much. it seems my problem is less to do with doing the actual math but rather deciphering what is being asked of me. the language of the science can sometimes be difficult for me.