 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 $1,x_1,x_2$. The necessary and sufficient conditions areUnder the assumed full rank, neither explanatory column is zero. The maximum-likelihood estimator is $\hat{\beta }=(X^{\mathsf{T}}X{)}^{-1}{X}^{\mathsf{T}}Y$, and the normal distribution of $Y$ impliesThus the three coefficient estimators are independent normal random variables, not merely uncorrelated. This is a sampling-distribution statement with the true ${\sigma }^2$ fixed.