Vector~~~~~

Sol
CosB=Cos60度=1/2=(1+1/9-OQ^2)/(2*1*1/3)
2*(1+1/9-OQ^2)=2*1*1/3
18*(1+1/9-OQ^2)=9*2*1/3
18+2-18OQ^2=6
18OQ^2=14
OQ^2=7/9
OQ=soqrt(7)/3=0.881917
Cos角BOQ=(0.881917^2+1-1/9)/(2*1*0.881917)
=.9449112
角BOQ=19.1066度
角BOQ=60度-2*19.1066度=21.7868度=22度

AB = b - a
OP = OA + AP = OA + AB/3 = a + (b-a)/3 = 2a/3 + b/3.
OQ = OA + OQ = OA + 2AB/3 = a + 2(b -a)/3 = a/3 + 2b/3.
OP . OP = (2a/3 + b/3) . ( 2a/3 + b/3) = 4/9 a . a + 4/9 a. b + 1/9 b . b
= 4/9 + 2/9 + 1/9 = 7/9, so magnitude of OP = (sqrt 7)/3
OQ . OQ = ( a/3 + 2b/3) . ( a/3 + 2b/3) = 1/9 a. a + 4/9 a. b + 4/9 b. b
= 1/9 + 2/9 + 4/9 = 7/9, so magnitude of OQ = (sqrt 7)/3
OP . OQ = ( 2a/3 + b/3) . (a/3 + 2b/3) = 2/9 a. a + 5/9 a. b + 2/9 b. b
= 2/9 + 5/18 + 2/9 = 13/18.
so 13/18 = magnitude of OP x magnitude of OQ cos (angle POQ)
= 7/9 cos (angle POQ)
so angle POQ = arc cos (13/14) = 21.78 = 22 degree.

Note: a . b = 1 x 1 x cos 60 = 1/2. a. a = 1 x 1 x cos 0 = 1 = b . b