(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

aortiH 2021-09-04 Answered

(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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Expert Answer

gwibdaithq
Answered 2021-09-05 Author has 6450 answers

(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) \(\displaystyle{\left|{u}\right|}={\left|{v}\right|}={1}{\left|{u}-{v}\right|}={2}\)
\(\displaystyle{\left|{u}-{v}\right|}^{{2}}+{\left|{u}+{v}\right|}^{{2}}={2}{\left|{u}\right|}^{{2}}+{2}{\left|{v}\right|}^{{2}}\)
\(\displaystyle\Rightarrow{u}+{\left|{u}+{v}\right|}^{{2}}={2}+{2}\)
\(\displaystyle\Rightarrow{\left|{u}+{v}\right|}^{{2}}={0}\)
\(\displaystyle\Rightarrow{\left|{u}+{v}\right|}={0}\)
\(\displaystyle\Rightarrow{u}+{v}={0}\)
\(\displaystyle\Rightarrow{u}=-{v}\)

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