※ 引述《YAchine (牙籤)》之銘言：
: Z1,Z2屬於C
: |Z1|=|Z1+Z2|=3,|Z1-Z2|=3(3)^0.5
: __ __
: 求log(3)|(Z1Z2)^2000+(Z1Z2)^2000|=
: 目前只能確定|Z2|=3
: 可利用題目給的條件建構平行四邊形 角度是60 120
: 不知道是不是要從隸美佛切入?
(Z1+Z2)˙(Z1-Z2) = 0 = Z1^2 - Z1Z2 + Z2Z1 - Z2^2 = |Z1|^2 - |Z2|^2
=> |Z2| = 3
(Z1+Z2)˙(Z1+Z2) = |Z1+Z2|^2*cos0°= 3^2 = Z1^2 + Z1Z2 + Z2Z1 + Z2^2
= |Z1|^2 + 2|Z1||Z2|cosθ + |Z2|^2
=> cosθ = -1/2 => θ=120°
=> 假設 Z1 = 3 + 0i => Z2 = 3/2 + 3√3/2 i
__ __
log(3)|(Z1Z2)^2000+(Z1Z2)^2000|
____
= log(3)|(Z1Z2)^2000+(Z1Z2)^2000|
= log(3)|[3^2(1/2 - √3/2 i)]^2000+[3^2(1/2 + √3/2 i)]^2000|
= log(3)|[3^2(cos60°- sin60°i)]^2000+[3^2(cos60°- sin60°i)]^2000|
= log(3)|3^4000(cos12000°- sin12000°i)+3^4000(cos12000°- sin12000°i)|
= log(3)|3^4000*cos120°+3^4000*cos120°|
= log(3)3^4000
= 4000
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