We will study the notion of right distributive ringoids over a field which are neither rings, semi-rings, semi-hyperrings nor near-rings. Matrices over ringoids are defined, and new concepts such as top-row-determinate and down-row-determinate related to 2 × 2 matrices over a ringoid are introduced. Moreover, we investigate the notions of the (strongly, (very-) weak) orthogonality of vectors over a ringoid. Beside, we discuss the notion of incident vectors and define the concept of α − K-sphere on a ringoid, where K is a field and investigate some of their properties. Finally, we show that in a commutative ringoid all of the vectors are strongly orthogonal.