uzasadnic , że równosc
\(\displaystyle{ \left| z_{1} + z_{2} \right| ^{2} + ft| z_{1} - z_{2} \right| ^{2} = 2 ( ft| z _{1} \right| ^{2} + ft| z_{2} \right| ^{2} )}\)
jest prawdziwa dla dowolnych liczb \(\displaystyle{ z_{1} , z_{2} liczb zespolonych}\)
uzasadnic
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scyth
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uzasadnic
\(\displaystyle{ |z_1+z_2|=\sqrt{[Re(z_1)+Re(z_2)]^2+[Im(z_1)+Im(z_2)]^2} \\
|z_1-z_2|=\sqrt{[Re(z_1)-Re(z_2)]^2+[Im(z_1)-Im(z_2)]^2} \\
|z_1+z_2|^2+|z_1-z_2|^2=[Re(z_1)+Re(z_2)]^2+[Im(z_1)+Im(z_2)]^2 + [Re(z_1)-Re(z_2)]^2+[Im(z_1)-Im(z_2)]^2 =\\=
[Re(z_1)]^2+2Re(z_1)Re(z_2)+[Re(z_2)]^2+[Im(z_1)]^2+2Im(z_1)Im(z_2)+[Im(z_2)]^2 +
[Re(z_1)]^2-2Re(z_1)Re(z_2)+[Re(z_2)]^2+[Im(z_1)]^2-2Im(z_1)Im(z_2)+[Im(z_2)]^2
=\\= 2[Re(z_1)]^2+2[Re(z_2)]^2+2[Im(z_1)]^2+2[Im(z_2)]^2=\\=
2(|z_1|^2+|z_2|^2)}\)
|z_1-z_2|=\sqrt{[Re(z_1)-Re(z_2)]^2+[Im(z_1)-Im(z_2)]^2} \\
|z_1+z_2|^2+|z_1-z_2|^2=[Re(z_1)+Re(z_2)]^2+[Im(z_1)+Im(z_2)]^2 + [Re(z_1)-Re(z_2)]^2+[Im(z_1)-Im(z_2)]^2 =\\=
[Re(z_1)]^2+2Re(z_1)Re(z_2)+[Re(z_2)]^2+[Im(z_1)]^2+2Im(z_1)Im(z_2)+[Im(z_2)]^2 +
[Re(z_1)]^2-2Re(z_1)Re(z_2)+[Re(z_2)]^2+[Im(z_1)]^2-2Im(z_1)Im(z_2)+[Im(z_2)]^2
=\\= 2[Re(z_1)]^2+2[Re(z_2)]^2+2[Im(z_1)]^2+2[Im(z_2)]^2=\\=
2(|z_1|^2+|z_2|^2)}\)