  In this paper, we aim to introduce and characterize the concept of numerical
radius orthogonality of operators on a complex Hilbert space $\mathcal{H}$
which are bounded with respect to the semi-norm induced by a positive operator
$A$ on $\mathcal{H}$. Moreover, a characterization of the $A$-numerical radius
parallelism for $A$-rank one operators is proved. As applications of the
obtained results, we obtain some $\mathbb{A}$-numerical radius inequalities of
operator matrices where $\mathbb{A}$ is the operator diagonal matrix with
diagonal entries are positive operator $A$. Some other related results are also
investigated.
