+226
Let z and w be complex numbers satisfying |z|=5, |w|=2 and z¯w=6+8i. Find in the numbers |z+w|2,|zw|2,|z−w|2,|zw|2
+118703 I really do not know what I am doing here. (I have forgotten it)
However.
|z|=5soz=5eθi|w|=2sow=2eαi|ˉw|=2soˉw=2e−αi zw=5eθi∗2eαi=10e(θ+α)i|zw|2=100|zw|2=254=6.25 zˉw=5eθi∗2e−αi=10e(θ−α)i=6+8iθ−α=atan(86)=atan(1.˙3)zˉw=6+8i=10e(atan(1.˙3))i
I think that is right as far as it goes but it does not answer all your questions.
LaTex:
|z|=5\;\;\quad so\;\; \qquad z=5e^{\theta i}\\
|w|=2\;\;\quad so\;\; \qquad w=2e^{\alpha i}\\
|\bar w|=2\;\;\quad so\;\; \qquad \bar w=2e^{-\alpha i}\\~\\
zw=5e^{\theta i}*2e^{\alpha i}=10e^{(\theta+\alpha)i}\\
|zw|^2=100\\
\left| \frac{z}{w} \right|^2=\frac{25}{4}=6.25\\~\\
z\bar w=5e^{\theta i}*2e^{-\alpha i}=10e^{(\theta-\alpha)i}=6+8i\\
\theta-\alpha=atan(\frac{8}{6})=atan(1.\dot3)\\
z\bar w=6+8i=10e^{(atan(1.\dot 3))i}
