Question

一道简单的向量题

已知A(-3,0),B(0,√3),O为坐标原点,点C在第二象限,且∠AOC=60°,向量OC=λ向量OA+向量OB,则实数λ的值是多少?

需要解题过程

Answer 1

OC = λOA + OB
= λ(-3,0) + (0,√3)
= (-3λ,√3)

OA . OC = (-3,0). (-3λ,√3)
= 9λ ----(1)
also
OA . OC = |OA||OC| cos∠AOC
= 3. [√(9λ^2 + 3 )] .1/2 --(2)

(1)=(2)
6λ = √(9λ^2 + 3 )
36λ^2 = 9λ^2 + 3
λ^2 = 1/9
λ = 1/3 or -1/3 (rejected)

As C is 第二象限, => λ = 1/3

Answer 2

OC=λOA+OB=(-3λ,√3)
cos∠AOC=OA*OC/(|OA|*|OC|)=9λ/(3√(9λ^2+3)=1/2
λ=±1/3
因为C在第二象限，所以λ=1/3