Eigenvectors of covariance matrix of residuals are orthogonal to the projection?

Question

Let $X \in \mathbb{R}^{t \times n}$ denote a data matrix. Let $W$ be a $t \times k$ matrix whose columns form an orthonormal basis of $\mathbb{R}^k$ where $k < n$. Let $R$ be the matrix of residuals such that $R = X - XWW^T$. Let $\Sigma_R$ be the covariance matrix of the residuals with eigenvalues $\lambda_1, \dots, \lambda_n$ and corresponding eigenvectors $v_1, \dots, v_n$. Is there a way to show that $W$ and $v_i$ are orthogonal $\forall \lambda_i > 0$? I.e. $W^T v_i = 0 \quad \forall \lambda_i > 0$?