Given $p \times q=g$ and $q \times g=p$ explain why $q$ is a unit vector

Question

I'm given $p \times q=g$ and $q \times g=p$ explain why $q$ is a unit vector, where $p, q, g$ are nonzero vectors and I have to explain why $q$ must be a unit vector. I know that the vectors are mutually perpendicular but it doesn't make sense to me as to why $q$ must be a unit vector. I tried using the fact they were all mutually perpendicular, i.e. $a \cdot b=a \cdot c = b \cdot c= 0$ but this didn't help either.

Answer 1

If you know the vectors are perpendicular, you have $|p||q|=|g|$ and $|q||g|=|p|$

Answer 2

We have $$|g|=|p\times q|=|p||q|\sin\frac\pi2\tag1$$ $$|p|=|q\times g|=|q||g|\sin\frac\pi2\tag2$$ so substituting $(2)$ into $(1)$ gives $$|g|=|q|^2|g|\sin^2\frac\pi2=|q|^2|g|\implies |g|(1-|q|^2)=0$$ so either $|g|=0$ which is impossible or $1-|q|^2=0\implies |q|=1$