Question

已知|向量a|=|向量b|=2，且a+b的夹角与a的夹角与a-b与a的夹角相等，求a与b的夹角。谢谢大家

Answer 1

let a+b的夹角与a的夹角 = x
a与b的夹角 = y
(a+b).a =|a|^2 + |a||b|cosx = 4+4cosy =|a+b||a|cosx
=>|a+b| = 2+2cosy (1)
(a-b).a =|a|^2 - |a||b|cosx = 4-4cosy =|a-b||a|cosx
=> |a-b| = 2-2cosy (2)
(1) +(2)
|a+b| +|a-b| = 4
consider
|a+b|^2 = |a|^2 +|b|^2 + 2|a||b|cosy
= 8 + 8cosy
= 16(1+cosy)/2
=16[cos(y/2)]^2
|a+b| = 4cos(y/2)
|a-b|^2 = |a|^2 +|b|^2 - 2|a||b|cosy
= 8 - 8cosy
= 16(1-cosy)/2
= 16(sin(y/2))^2
|a-b| = 4siny/2

|a+b| +|a-b| = 4
4cos(y/2) + 4sin(y/2) =4
cos(y/2) +sin(y/2)=1
(cos(y/2))^2 = 1+ (sin(y/2))^2 + 2sin(y/2)
[sin(y/2)]^2 + [sin(y/2)] =0
y =0 or y = 3π/4
a与b的夹角 = 0 or 3π/4