Question

向量a=(cosx,sinx) b=(cosy,siny) 为什么它们的夹角是y-x

Answer 1

设夹角z,

cosz=a∙b/(|a||b|)-----------------------------------------------(1)

|a|=√[(cos(x))^2+(sin(x))^2]=1---------------------------------(2)

|b|=√[(cos(y))^2+(sin(y))^2]=1---------------------------------(3)

a∙b=cosx cosy+sinx siny=cos(x-y)=cos(y-x)------------(4)

从(1),(2),(3),(4), cosz=cos(y-x)--------------------------(5)

z=y-x

|a+b|=√(a+b)∙(a+b)=√{a∙a)+2(a∙b)+b∙b}---------------(6)

从(2),(3),(4),

|a+b|=√(2+2cos(x-y))

同理,|a-b|=√(2-2cos(x-y))