Paper 1, Section I, A

Vectors and Matrices | Part IA, 2017

Consider zCz \in \mathbb{C} with z=1|z|=1 and argz=θ\arg z=\theta, where θ[0,π)\theta \in[0, \pi).

(a) Prove algebraically that the modulus of 1+z1+z is 2cos12θ2 \cos \frac{1}{2} \theta and that the argument is 12θ\frac{1}{2} \theta. Obtain these results geometrically using the Argand diagram.

(b) Obtain corresponding results algebraically and geometrically for 1z1-z.

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