Given three complex numbers $|Z_1|= 2 , |Z_2|= 3, |Z_3| = 4$ find the maximum value of $$|Z_1-Z_2|^2 +|Z_2-Z_3|^2+|Z_3-Z_1|^2$$
If we treat them as three vectors $a, b, c$ centred at zero the above expression becomes $$2(|a|^2+|b|^2+|c|^2)-2(a\cdot b + b\cdot c + c\cdot a)$$
I've been unsuccessful trying to find the minimum value of $a\cdot b + b\cdot c + c\cdot a$.