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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 e^{jθ}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 = e^{jθ}

j^{2}= -1

3. The attempt at a solution

so z(e^{jθ})

= (a + jb)(cosθ + jsinθ)

= acosθ + (asinθ + bcosθ)j - bsinθ

= (acosθ - bsinθ) + (asinθ + bcosθ)j

Now I'm supposed to show that the above equation isa 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, L_{o}:

L_{o}^{2}= a^{2}+ b^{2}

L_{o}= √(a^{2}+b^{2})

Length of new z, L_{1}:

L_{1}^{2}= (acosθ - bsinθ)^{2}+ (asinθ + bcosθ)^{2}

L_{1}^{2}= a^{2}+ b^{2}

L_{1}= √(a^{2}+b^{2})

Therefore L_{o}= L_{1}

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!

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# Help with Complex Exponentials

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