Question

向量问题 已知|OA|=2，|OB|=√3，∠AOB=120°，点C在∠AOB内

已知|OA|=2，|OB|=√3，∠AOB=120°，点C在∠AOB内，且∠AOC=30°，设OC=mOA+nOB(m,n∈R),则m/n=？

Answer 1

以下(a.b)表示a点乘b.
= = = = = = = = =

解：因为 点C在∠AOB内,
且 ∠AOB=120°, ∠AOC=30°,
所以 ∠BOC=90°.
设 |OC|= x>0 .
因为 |OA|= 2, |OB|= √3,
所以 (OA.OB)= |OA| *|OB| *cos ∠AOB
= 2√3 cos 120°
= -√3 .
(OC.OA)= |OC| *|OA| *cos ∠AOC
= 2x cos 30°
= (√3)x .
(OC.OB)= |OC| *|OA| *cos ∠BOC
= (√3)x cos 90°
=0 .
又因为 OC= m OA +n OB,
所以 (OC.OA)= m OA^2 +n (OB.OA)
= 4m -(√3)n ,
(OC.OB)= m (OA.OB) +n OB^2
= -(√3)m +3n.
所以 4m -(√3)n = (√3)x , (1)
-(√3)m +3n =0. (2)
解得 m= [ -(√3)/3 ]x,
n= (1/3) x.
所以 m/n = √3 .

= = = = = = = = =
计算可能有误，你最好检查一下。