Earlier:
a.a = (2u + 3v).(2u + 3v) = 4u.u + 9v.v + 6u.v + 6v.u = 4u.u + 9v.v + 12u.v
u.u = 1
v.v = 1
u.v = cos(60) = 1/2
Therefore, a.a = 4 + 9 + 12/2 = 19
a.a = |a|^2, so the length of a is sqrt(a.a) = sqrt(19)
You went wrong with the dot products, substituting 2 and 3 for u and v. If there is still an issue, could you explain why you did that?
Edit: When writing vector symbols, I underline them, so that they do not get confused with scalars. I should have done that when I typed them too.