If $\vec{a} \times \vec{b}=\vec{c}$ and $\vec{b} \times \vec{c}=\vec{a}$ then

$|\vec{a}|=|\vec{c}|$

$\vec{a}(\vec{a} \times \vec{b})=\vec{a} . \vec{c}=0 $ Also $\vec{b}(\vec{a} \times \vec{b})=\vec{b} . \vec{c}=0$, and $\vec{b}(\vec{b} \times \vec{c})=\vec{a} . \vec{b}=0$

$\Rightarrow \vec{a}, \vec{b}, \vec{c}$ are mutually perpendicular vectors.

$\Rightarrow |\vec{a} \times \vec{b}|=|\vec{a}||\vec{b}|=|\vec{c}|$ and $|\vec{b}||\vec{c}|=|\vec{a}| \Rightarrow|\vec{b}|=1 \Rightarrow|\vec{a}|=|\vec{c}|$

Hence (3) is correct answer.