On zeros of a system of quadratic forms in 3 variables

Let k be a field of characteristic distinct from 2, f1, f2, f3 quadratic forms in 3 variables over k. We prove that these forms have a common zero over k if and only if the quadratic form u1f1+u2f2+u3f3 is isotropic over the field k(u1,u2,u3) and the cubic form det(t1f1+t2f2+t3f3) in variables t1, t2, t3 is isotropic over k. We also consider a similar question for n quadratic forms in 3 variables where n≥4.