Let $\mathcal F$, $\mathcal G$ be vector bundles of rank $n$, $m$. Then how to compute $det(\mathcal F \otimes \mathcal G)$ and $det(\mathcal F \oplus \mathcal G)$?
I know the answer is $det(\mathcal F \otimes \mathcal G)=det(\mathcal F)^{\otimes m}\otimes det(\mathcal G)^{\otimes n}$ and $det(\mathcal F \oplus \mathcal G)=det(\mathcal F)\otimes det(\mathcal G)$. I remember them as matrixes, but I don't know how to prove them.
Thanks in advance.