已知向量m=(sinx,-1)向量n=(根号3cosx,-12)​若x=π/6,求向量m、n夹角θ

题目

已知向量m=(sinx,-1)向量n=(根号3cosx,-12)​若x=π/6,求向量m、n夹角θ

答案

m×n=|m||n|cosθ
m×n=sinx×√3cosx+(-1)×(-1/2)
=√3sinxcosx+1/2
=√3sin2x/2+1/2
=√3sinπ/3+1/2
=√3×√3/2+1/2
=2
|m|=√(sinx^2+1)=√(5/4)
|n|=√(3cosx^2+1/4)=√(5/2)
|m||n|=5/2√2=5√2/4
∴cosθ=2/(5√2/4)=4√2/5
∴θ=arccos4√2/5