Question

已知向量a（ sinx， cosx） b （sinx+cosx, sinx-cosx ）

求|a+b|取值范围

Answer 1

(a+b)=(2sinx+cosx, sinx)
|a+b|^2 =(2sinx+cosx)^2+(sinx)^2
= 4(sinx)^2+ 4sinxcosx +1
= 2(1- cos2x) + 2sin2x +1
= 2(sin2x-cos2x)+3
=2√2sin(2x-π/4)+3

3-2√2 <=|a+b|^2 <=3+2√2
√(3-2√2) <=|a+b| <=√(3+2√2)