Consider the linear regression model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\varepsilon\), where the regressors have been coded so that

\[
\sum_{i=1}^{n} x_{i 1}=\sum_{i=1}^{n} x_{i 2}=0 \quad \text { and } \quad \sum_{i=1}^{n} x_{i 1}^{2}=\sum_{i=1}^{n} x_{i 2}^{2}=n
\]

a. Show that an orthogonal design ( \(\mathbf{X}^{\prime} \mathbf{X}\) diagonal) minimizes the variance of \(\hat{\beta}_{1}\) and \(\hat{\beta}_{2}\).
b. Show that any design for fitting this first-order model that is orthogonal is also rotatable.