Consider the linear model

$$Y_{i}=\beta_{0}+\beta_{1} x_{i 1}+\beta_{2} x_{i 2}+\varepsilon_{i}$$

for $i=1,2, \ldots, n$, where the $\varepsilon_{i}$ are independent and identically distributed with $N\left(0, \sigma^{2}\right)$ distribution. What does it mean for the pair $\beta_{1}$ and $\beta_{2}$ to be orthogonal? What does it mean for all the three parameters $\beta_{0}, \beta_{1}$ and $\beta_{2}$ to be mutually orthogonal? Give necessary and sufficient conditions on $\left(x_{i 1}\right)_{i=1}^{n},\left(x_{i 2}\right)_{i=1}^{n}$ so that $\beta_{0}, \beta_{1}$ and $\beta_{2}$ are mutually orthogonal. If $\beta_{0}, \beta_{1}, \beta_{2}$ are mutually orthogonal, find the joint distribution of the corresponding maximum likelihood estimators $\hat{\beta}_{0}, \hat{\beta}_{1}$ and $\hat{\beta}_{2}$.