Dot product between 2 random vectors

Question

I have 2 independent 2-dimenstional random vectors $A$ and $B$.

$$ A = [a_1, a_2] $$ and $$ B= [b_1, b_2] $$

The variance of the elements of A are identical ($Var[a_1] = Var[a_2] = \sigma^2_a$). The variance of the elements of B are also identical ($Var[b_1] = Var[b_2] = \sigma^2_b$), however they are correlated by $\rho$.

I need to take the dot product between A and B, and somehow isolate $\sigma^2_a$.

Could you please tell me if my reasoning is correct ?

\begin{align} Y &= A \cdot B \\[0.7em] & = a_1b_1 + a_2b_2 \\[0.7em] Var[Y] &= Var[a_1b_1] + Var[a_2b_2] + 2Cov(a_1b_1, a_2b_2) \\[0.7em] &= Var[a_1]Var[b_1] + Var[a_1]Var[b_1] + 2\sigma^2_a \ Cov(b_1, b_2) \\[0.7em] &= \sigma^2_a \left(Var[b_1] + Var[b_2] + 2Cov(b_1, b_2) \right) \\[0.7em] &= 2 \sigma^2_a \ \sigma^2_b(1+ \rho) \end{align}

Thanks,

Liam