Euclidean spaces

Britney is choosing a basis $\left\{\vec{e}_{1},\vec{e}_{2}\right\}$ of $V$ (two-dimensional vector space) and Allan gives her three real numbers $a_{11}$, $a_{22}$ and $a_{12}$. Britney checks if there is an inner product $\langle\cdot,\cdot\rangle$, such that $$\left\langle\vec{e}_{1},\vec{e}_{1}\right\rangle= a_{11}\\ \left\langle\vec{e}_{2},\vec{e}_{2}\right\rangle= a_{22}\\ \left\langle\vec{e}_{1},\vec{e}_{2}\right\rangle= a_{12}$$If there is such an inner product, then Britney gives Allan a candy.

Is she giving Allan a candy if his numbers are $a_{11} = 1$, $a_{22} = 2$, $a_{12} = -3$?