Hi,

I'm having trouble proving the fact that the counterclockwise rotation about the positive z-axis through an angle θ becomes

w_1= xcosθ－ysinθ

w_2= xsinθ+ycosθ

w_3= z

when X=(x,y,z) is rotated counterclockwise to obtain w=(w_1,w_2,w_3)

I understand how this would hold when the diameter of the base of the cone(in this case =2r) is set along the x-axis, and the angle between the x-axis and X is defined by φ.

Explanation:

As x=rcosφ, y=rsinφ,

x=cos(θ+φ), y=sin(θ+φ).

From the addition theorem of trigonometric functions,

w_1= xcosθ－ysinθ

w_2= xsinθ+ycosθ

w_3= z

But wouldn't w_1 and w_2 be defined differently if the diameter of the base of the cone is on the y-axis? Wouldn't x= rsinθ,y=rcosθ?

Please help!