= Solution
Partition the <design matrix> accordingly as $X=(X_1\ \cdots\ X_k)$. The blocks $\beta_i,\beta_j$ are <orthogonal> when $X_i^{\mathsf T}X_j=0$, meaning that their column spaces are <orthogonal vectors>. Mutual <orthogonality> means that this holds for every pair of distinct blocks, so $X^{\mathsf T}X$ is block diagonal.
For the three scalar coefficients, the corresponding columns are $\mathbf1,x_1,x_2$. The necessary and sufficient conditions are
$$
\boxed{\sum_{i=1}^n x_{i1}=0,\qquad\sum_{i=1}^n x_{i2}=0,\qquad\sum_{i=1}^n x_{i1}x_{i2}=0.}
$$
Under the assumed full rank, neither explanatory column is zero. The <maximum-likelihood estimator> is $\widehat\beta=(X^{\mathsf T}X)^{-1}X^{\mathsf T}Y$, and the <normal distribution> of $Y$ implies
$$
\widehat\beta\sim N_3\!\left(\beta,\sigma^2\operatorname{diag}\left(\frac1n,\frac1{\sum_i x_{i1}^2},\frac1{\sum_i x_{i2}^2}\right)\right).
$$
Thus \b[the three coefficient estimators are independent normal random variables], not merely uncorrelated. This is a sampling-distribution statement with the true $\sigma^2$ fixed.
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