Question #74448

Let $\vec{a}$ and $\vec{b}$ be two mutually orthogonal unit vectors, and let $\vec{c} = \vec{a} + \vec{b} - \vec{a} \times \vec{b}$. A point $\vec{r}_0$ is the unique intersection of the three planes given by the equations: $\vec{r} \cdot \vec{a} = t, \quad \vec{r} \cdot \vec{b} = 2t - 1, \quad \vec{r} \cdot \vec{c} = 2t^2 + 3t - 2$ where $t \in \mathbb{R}$. It is given that $\vec{r}_0$ satisfies the following two conditions: (i) $\vec{r}_0$ lies in the region containing the transverse axis bounded by the asymptotic planes of the hyperbolic cylinder $(\vec{r} \cdot \vec{a})^2 - (\vec{r} \cdot \vec{b})^2 = 1$. (ii) The scalar projection of $\vec{r}_0$ along the vector $\vec{a} \times \vec{b}$ is strictly greater than $\frac{1}{2}$.

Find the set of all possible values of $t$.