1. The problem statement, all variables and given/known data Hello all, I'm currently reading through "Vibrations and Waves" by A.P French. I'm not very used to these type of books, so I could use a little bit of help right now: Show that the multiplication of any complex number z by ejθ is describable, in geometrical terms, as a positive rotation through the angle θ of the vector by which z is represented, without any alteration of its length. 2. Relevant equations z = a + jb j = an instruction to perform a counterclockwise rotation of 90 degrees upon whatever it precedes. cosθ + jsinθ = ejθ j2 = -1 3. The attempt at a solution so z(ejθ) = (a + jb)(cosθ + jsinθ) = acosθ + (asinθ + bcosθ)j - bsinθ = (acosθ - bsinθ) + (asinθ + bcosθ)j Now I'm supposed to show that the above equation is a positive rotation through the angle θ of the vector by which z is represented, without any alteration of its length. First, I'm going to prove that the length was not altered: Original length of z, Lo: Lo2 = a2 + b2 Lo = √(a2+b2) Length of new z, L1: L12 = (acosθ - bsinθ)2 + (asinθ + bcosθ)2 L12 = a2 + b2 L1 = √(a2+b2) Therefore Lo = L1 Now, How do I prove the rotation? I'm not too strong with trig, maybe that's the problem? Can somebody help me out? Thank you!