a=(λ,2),b=(3,5)且a与b夹角θ∈【π/3,2π/3】则λ取值范围

题目

a=(λ,2),b=(3,5)且a与b夹角θ∈【π/3,2π/3】则λ取值范围

答案

设b=(3,5)与X轴夹角为x
x=arctan(5/3)
则tan[arctan(5/3)+π/3]={tan[arctan(5/3)]+tan(π/3）}/{1-tan[arctan(5/3)]tan(π/3）}
=[5/3+根号3]/[1-5/3*根号3]=-(30+17根号3)/33
λ=-60+34根号3
tan[arctan(5/3)+2π/3]={tan[arctan(5/3)]+tan(2π/3）}/{1-tan[arctan(5/3)]tan(2π/3）}
=[5/3-根号3]/[1+5/3*根号3]=(-30+17根号3)/33
λ=-60-34根号3
所以λ∈【-60-34根号3,-60+34根号3】