向量a=(cosx,sinx) b=(cosy,siny) 为什么它们的夹角是y-x
3个回答
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))
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