Compute ||a|| given that e/ā=5/4+i/4 and ea=10+2i
I found sqrt(13) as answer, but the solution says its 2sqrt(2), am I doing something wrong?
Thanks a lot!
Compute ||a|| given that e/ā=5/4+i/4 and ea=10+2i
I found sqrt(13) as answer, but the solution says its 2sqrt(2), am I doing something wrong?
Thanks a lot!
$e\cdot a=10+2i$
$\frac e {\bar a} =\frac{5+i}4$
On division $a\cdot \bar a=\frac{4(10+2i)}{5+i}=8\cdot \frac{5+i}{5+i}=8$
If $a=x+iy, \bar a=x-iy,$ so, $a\cdot \bar a=(x+iy)(x-iy)=x^2+y^2=\mid a\mid ^2$
So, $\mid a\mid ^2=8\implies |a|=2\sqrt2 $
As $e\cdot a=10+2i, \mid e\cdot a \mid=\sqrt{10^2+2^2}=\sqrt {104}$
As $\mid e\cdot a \mid= \mid e\mid\cdot \mid a \mid, $
so, $\mid e\mid\cdot \mid a \mid= \sqrt {104}$
but, $\mid a \mid=\sqrt 8,$ so, $\mid e\mid=\frac{\sqrt{104}}{\sqrt{8}}=\sqrt{13}$