I am struggling to understand the 3D rotation. 
The initial object position is A(x1,y1,z1). We need rotate it into A′(x2,y2,z2) about z-axis by angle θ.And we take P be the any point in z-axis. It makes ∠APS=ϕ.
Now we find cosϕ=PS/OP=x1/r⇒x1=rcosϕ. But we know OQ=x1 but here slides take PS=x1 and match the answer.
My question is slides has wrong or my concepts is wrong? If both are wrong so how we find 3D rotation about any coordinate axis?
The dotted vector which ends to A(x1,y1,z1) and has length r does not start from the origin, it starts from P. The drawing visualizes rotation around Z-axis. It doesn't affect the Z-coordinate of the rotated point. Everything should be calculated with the XY-plane projection of the point. Here a copy of XY plane is placed to point P.