# (a)Show that for all complex numbers z and w. |z-w|^2+|z+w|^2=2|z|^2+2|w|^2. (b) Let u,v be complex numbers such that |u|=|v|=1 and |u-v|=2

(a)Show that for all complex numbers z and w
$$\displaystyle{\left|{z}-{w}\right|}^{{2}}+{\left|{z}+{w}\right|}^{{2}}={2}{\left|{z}\right|}^{{2}}+{2}{\left|{w}\right|}^{{2}}$$
(b) Let u,v be complex numbers such that $$\displaystyle{\left|{u}\right|}={\left|{v}\right|}={1}{\quad\text{and}\quad}{\left|{u}-{v}\right|}={2}$$ Use part (a) to express u in terms of v.

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(a) $$z,w \in ะก$$
$$\displaystyle{\left|{z}-{w}\right|}^{{2}}+{\left|{z}+{w}\right|}^{{2}}={2}{\left|{z}\right|}^{{2}}+{2}{\left|{w}\right|}^{{2}}$$
Step 2
$$\displaystyle{\left|{z}-{w}\right|}^{{2}}+{\left|{z}+{w}\right|}^{{2}}={\left({z}-{w}\right)}{\left(\overline{{{z}-{w}}}\right)}+{\left({z}+{w}\right)}{\left(\overline{{{z}+{w}}}\right)}$$
$$\displaystyle={\left({z}-{w}\right)}{\left(\overline{{{z}}}-\overline{{{w}}}\right)}+{\left({z}+{w}\right)}{\left(\overline{{{z}}}+\overline{{{w}}}\right)}$$
$$\displaystyle={z}\overline{{{z}}}-{z}\overline{{{w}}}-{w}\overline{{{z}}}+{w}\overline{{{w}}}$$
$$\displaystyle+{z}\overline{{{z}}}+{z}\overline{{{w}}}+\overline{{{w}}}{z}+{w}\overline{{{w}}}$$
$$\displaystyle\Rightarrow{\left|{z}-{w}\right|}^{{2}}+{\left|{z}+{w}\right|}^{{2}}={2}{\left|{z}\right|}^{{2}}+{2}{\left|{w}\right|}^{{2}}$$
(b) $$\displaystyle{\left|{u}\right|}={\left|{v}\right|}={1}{\left|{u}-{v}\right|}={2}$$
$$\displaystyle{\left|{u}-{v}\right|}^{{2}}+{\left|{u}+{v}\right|}^{{2}}={2}{\left|{u}\right|}^{{2}}+{2}{\left|{v}\right|}^{{2}}$$
$$\displaystyle\Rightarrow{u}+{\left|{u}+{v}\right|}^{{2}}={2}+{2}$$
$$\displaystyle\Rightarrow{\left|{u}+{v}\right|}^{{2}}={0}$$
$$\displaystyle\Rightarrow{\left|{u}+{v}\right|}={0}$$
$$\displaystyle\Rightarrow{u}+{v}={0}$$
$$\displaystyle\Rightarrow{u}=-{v}$$