🧣Vector Proofs of Rhombus Orthogonality (VP-RO)

The relationship between the diagonals of a parallelogram—representing the sum and difference of its spanning sides—reveals that they are orthogonal only when the sides have equal magnitudes. The sources explain that the algebraic identity governing these diagonals ensures their dot product reaches zero only when the side lengths are identical, a condition that transforms a general parallelogram into a rhombus. While unequal sides result in oblique intersections, matching side lengths force a perpendicular crossing, which can be visually demonstrated as the intersection angle hits exactly ninety degrees. This geometric principle serves as a practical magnitude check, as it confirms that the only way for these internal lines to be perpendicular is for the outer sides to be of equal length.