1. circumscribed circle: the center of a triangle circumscribed circle, which is the intersection point of the perpendicular lines of three sides.

2. Sine theorem: a/sinA=b/sinB=c/sinC=2R(R is the radius of the circumscribed circle of a triangle).

(1) According to the cosine theorem: cos ∠ BAC = cos120 = (ab&; sup2+AC & amp； Sup2 BC & sup2)/2× AB× AC

Solution: BC=√7

According to sine theorem: BC/sin∠BAC=2R(R is the radius of the circumscribed circle of △ABC).

∴R=√2 1/3

|AO| is the radius of the circumscribed circle of △ABC, |AO|=R=√2 1/3.

(2) As shown in the figure below, take the straight line of AB as the X axis and the straight line of the perpendicular line of AB as the Y axis, and establish a plane rectangular coordinate system.

Then a (- 1, 0) o (0,2 √ 3/3) b (1,0) c (-3/2, √ 3/2).

Vector ao = (1, 2 √ 3/3)

Vector a= vector ab = (2 2,0)

Vector b= vector AC=(- 1/2, √3/2)

∫ vector AO=λ 1×a+λ2×b

That is, (1, 2 √ 3/3) = λ1(2,0)+λ 2 (-1/2, √3/2).

∴ Available equation:1= 2λ1-(1/2) λ 2①

2√3/3=? √3/2λ2? ②

Solution: λ 1=5/6? λ2=4/3

So λ1+λ 2 = 5/6+4/3 =13/6.