Question about absolute value of the addition of vectors

Question

I would say the answer is "The angle between A and B is 120" because it |A| describes the L2 norm and every other option could not yield a correct result 1

Am I right? If so, explain why. The friend that asked this said it's wrong and it was because of a Chegg study post so I want to know why he was right/wrong.

Answer 1

$|A + B|^2 = |A|^2 + |B|^2 +2(A\cdot B)$ which yields $$ \begin{align} A\cdot B &= \frac{1}{2}\{|A + B|^2 - (|A|^2 + |B|^2)\}\\ &= -\frac{1}{2}|A|^2 \end{align} $$ Since $A\cdot B = |A|^2\cos\theta$, we have

1. $|A| = |B| = 0$ or
2. $|A| = |B| \not= 0$ and $\cos\theta = -1/2$, which gives one of the solutions $\theta = 120\deg$.

The problem missed case 1 though.