It is defined in such a way that for ${\displaystyle v\otimes w\in V\otimes W}$ we have ${\displaystyle q(v\otimes w)=q_{1}(v)q_{2}(w)}$. In particular, if we have diagonalizations of our quadratic forms (which is always possible when the characteristic is not 2) such that
${\displaystyle q_{1}\cong \langle a_{1},...,a_{n}\rangle }$
${\displaystyle q_{2}\cong \langle b_{1},...,b_{m}\rangle }$
${\displaystyle q_{1}\otimes q_{2}=q\cong \langle a_{1}b_{1},a_{1}b_{2},...a_{1}b_{m},a_{2}b_{1},...,a_{2}b_{m},...,a_{n}b_{1},...a_{n}b_{m}\rangle .}$