(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

aortiH

Answered question

2021-09-04

(a)Show that for all complex numbers z and w
|zw|2+|z+w|2=2|z|2+2|w|2
(b) Let u,v be complex numbers such that |u|=|v|=1and|uv|=2 Use part (a) to express u in terms of v.

Answer & Explanation

gwibdaithq

gwibdaithq

Skilled2021-09-05Added 84 answers

(a) z,wС
|zw|2+|z+w|2=2|z|2+2|w|2
Step 2
|zw|2+|z+w|2=(zw)(zw)+(z+w)(z+w)
=(zw)(zw)+(z+w)(z+w)
=zzzwwz+ww
+zz+zw+wz+ww
|zw|2+|z+w|2=2|z|2+2|w|2
(b) |u|=|v|=1|uv|=2
|uv|2+|u+v|2=2|u|2+2|v|2
u+|u+v|2=2+2
|u+v|2=0
|u+v|=0
u+v=0
u=v

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