|a|=lb|=|a+b|则向量a-b与b的夹角为

Question

|a|=lb|=|a+b|则向量a-b与b的夹角为

Answer 1

|a+b|^2=(a+b)^a^2+2ab+b^2.
=|a|^2+2a||b|cos+|b|^2
=|b|^2+|b|^2+2|b||b|coa .
∵|a|^2=|b|^2=|a+b|^2.
∴ 2|b|^2+2|b|^2cos=|b|^2.
2+2cos=1.
cos,a,b>=-1/2.
∴=120°
|a-b|^2=|a+b|^2-4ab.
=2|b|^2-2|b|^2cos.
=2|b|^2－2|b|^2*(-1/2）.
=3|b|^2.
∴|a-b|=√3|b|.
(a-b).b=(ab-b^2).
=(|a||b|cos-|b|^2.( |a|=|b|).
=|b|^2(1-cos.
=|b|^2(1+1/2).
=(3/2)|b|^2.
由向量公式,得：
cos=(a-b).b/(|a-b|.|b|).
=(3/2)|b|^2/(√3|b||b|).
=√3/2.
=30°.
∴向量a-b与向量b的夹角为30°.