I need some help here. It's hard to me to tackle this kind of question and I'm not used to write math proofs. I need to find values of $t$ that make $$\langle(x_1, x_2), (y_1, y_2)\rangle = x_1y_1 + tx_2y_2$$ an internal product in $\Bbb R^2$.
I have showed that for $3$ of the $4$ properties, t does not matter at all. But for the property that $$\langle u,u\rangle \gt 0, u \neq 0$$ I have $$x_1^2 +tx_2^2 \gt 0$$ and them $$t \gt -((x_1/x_2)^2)$$ The answer seems to be $t \gt 0$ and I'm lost in this. Could anyone give me the right direction to complete the proof?