Let a $\in \mathbb{R}$ and $q_{a}$, the quadratic forms defined by $q_{a}(x; y ; z)=a\left(x^{2}+y^{2}+z^{2}\right)+2 x z-2 x y+2 y z .$
I want to prove that there exists a same basis of $\mathbb{R}^{3}$ which is orthogonal to all the quadratic forms $q_{a}$.
I considered separately the quadratic forms $q_{0}(x, y, z)=2 x z-2 x y+2 y z$ and $q{\prime}(x ; y, z)=x^{2}+y^{2}+z^{2}$. Then i looked for an orthogonal basis of $q_{a}$ and tried to see if i could manipulate the vectors of this basis making it orhtogonal for the quadratic form $q{\prime}$. But it doesn't work.
Actually, i really really don't see how to solve the problem. Thanks in advance.