已知点A(3,根号3),O为坐标原点,点P(x,y)满足:根号3x-y≤0,x-根号3y+2≥0,y≥0,则向量OA*向量OP/|向量OP|的范围?

Question

已知点A(3,根号3),O为坐标原点,点P(x,y)满足:根号3x-y≤0,x-根号3y+2≥0,y≥0,
则向量OA*向量OP/|向量OP|的范围?

Answer 1

令所求值为m,m= OA • OP/| OA | =| OA |cos∠AOP=2√3 cos∠AOP,
因为∠AOP=[π/6 ,5π/6 ],所以当∠AOP=π/6 时,mmax=2√3 cos=π/6 =3；
当∠AOP=5π/6 时,mmin=2√3 cos5π/6 =-3．∴m= OA • OP/| OA | 的取值范围为[-3,3]．