Answer to Question #263288 in Algebra for Amana Ateba

Question

Let z, w ∈ C. Show |z + w| =

p

|z|

2 + 2Re(zw) + |w|

2 =

p

|z|

2 + 2|z||w| cos(θz − θw) + |w|

2 and

|z + w| = (|z| + |w|)

s

1 −



√

2 sin([θz−θw]/2)

|z|/|w|+

√

|w|/|z|

2

, if z 6= 0 and w 6= 0.

Answer 1

Let z, w ∈ C. Show

$|z + w| =\sqrt{|z|^2 + 2Re(z\overline{w}) + |w|^2} =\sqrt{|z|^2 + 2|z||w| cos(θ_z − θ_w) + |w|^2}$

if $z \neq 0 ,w \neq 0$ .

$|z + w|^2=(z+w)(\overline{z+w})=(z+w)(\overline{z}+\overline{w})=z\overline{z}+w\overline{w}+z\overline{w}+\overline{z}w=$

$=|z|^2+|w|^2+z\overline{w}+\overline{z}w=|z|^2+|w|^2+2Re(z\overline{w})$

$z\overline{w}=|z||w|(cos\theta_z+isin \theta_z)(cos\theta_w-isin \theta_w)$

$Re(z\overline{w})=cos\theta_z cos\theta_w+sin\theta_z sin\theta_w=|z||w| cos(θ_z − θ_w)$