设三角形OPQ的面积为S,已知OP向量·PQ向量=1.（1）若S∈（1/2,√3/2）,求向量OP与PQ的夹角θ的取值范围；（2）若S=3/4丨OP向量丨,求丨OQ向量丨的最小值.

设三角形OPQ的面积为S,已知OP向量·PQ向量=1.（1）若S∈（1/2,√3/2）,求向量OP与PQ的夹角θ的取值范围；（2）若S=3/4丨OP向量丨,求丨OQ向量丨的最小值.
设三角形OPQ的面积为S,已知OP向量·PQ向量=1.
（1）若S∈（1/2,√3/2）,求向量OP与PQ的夹角θ的取值范围；
（2）若S=3/4丨OP向量丨,求丨OQ向量丨的最小值.

设三角形OPQ的面积为S,已知OP向量·PQ向量=1.（1）若S∈（1/2,√3/2）,求向量OP与PQ的夹角θ的取值范围；（2）若S=3/4丨OP向量丨,求丨OQ向量丨的最小值.
（1）易得θ=180°-∠P
OP向量·PQ向量=丨OP向量丨*丨PQ向量丨*cosθ=1
s=1/2*丨OP向量丨*丨PQ向量丨*sinP=1/2*丨OP向量丨*丨PQ向量丨*sinθ
tanθ=2s∈（1,√3）,则θ∈（45°,60°）
（2）s=1/2*丨OP向量丨*丨PQ向量丨*sinθ=3/4丨OP向量丨
则丨PQ向量丨=3/(2sinθ)
tanθ=2s=3/2丨OP向量丨
由余弦定理 丨OQ向量丨^2=丨OP向量丨^2+丨PQ向量丨^2-2*丨OP向量丨*丨PQ向量丨*cosP
=丨OP向量丨^2+丨PQ向量丨^2+2*丨OP向量丨*丨PQ向量丨*cosθ
=9/(4tanθ^2)+9/(4sinθ^2)+2
=9/4(sinθ/cosθ)^2+9/4(cosθ/sinθ)^2+9/4+2
>=9/4*2*(sinθ/cosθ)*(cosθ/sinθ)+9/4+2=25/4
当且仅当sinθ/cosθ=cosθ/sinθ即θ=45°时取等号
(即a^2+b^2>=2ab当且仅当a=b时取等号)
所以丨OQ向量丨最小值为5/2

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