Determining the possible ratio of magnitude of two vectors A and B

Question

Let $\vec{A}$ and $\vec{B}$ be two non-null vectors such that $|$$\vec{A}$ + $\vec{B}$$|$ = $|$$\vec{A}$ - $\vec{2B}$$|$. The the value of $|$$\vec{A}$$|$/$|$$\vec{B}$$|$ may be:

(a) 1/4
(b) 1/8
(c) 1
(d) 2

Please give an explanation.

Answer 1

Squaring both sides we get $|\vec{A}|^2 + |\vec{B}|^2 + 2\vec{A} \cdot \vec{B} = |\vec{A}|^2 + 4|\vec{B}|^2 - 4\vec{A} \cdot \vec{B} $ and hence $|\vec{B}|^2 = 2 \vec{A}\cdot \vec{B}$. Thus $|\vec{A}|/|\vec{B}| = \frac{1}{2\cos\theta}$ where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$. Since $|\cos \theta|\leq 1$, out of the given answers, (c) and (d) are possible.