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JJHK
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Homework Statement
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.
Homework Equations
z = a + jb
j = an instruction to perform a counterclockwise rotation of 90 degrees upon whatever it precedes.
cosθ + jsinθ = ejθ
j2 = -1
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!