(1)
m.n = 1/2
(cos C/2,sin C/2).(cos C/2,-sin C/2) = 1/2
(cosC/2)^2 - (sinC/2)^2 = 1/2
cos2C = 1/2
2C = 60° or 300°
C=30° or 150°
(2)
c= 7/2 ,三角形ABC的面积S=3√3/2
To find:a+b
S = (1/2)absinC
ab =3√3/2
By cosine rule
c^2 = a^2+b^2 - 2abcosC
= (a+b)^2 - 2ab -2abcosC
if C = 30°
(7/2)^2 = (a+b)^2 - 2(3√3/2)(1+√3/2)
49/4 =(a+b)^2 -3√3(2+√3)/2
(a+b)^2 = 67/4 +3√3
a+b = √(67/4 +3√3)
if C= 150°
a+b = √(31/4 +3√3)