Diagonals Are Perpendicular Only In A Rhombus

A parallelogram's diagonals, represented by the vectors $\vec{v}+\vec{w}$ and $\vec{v}-\vec{w}$, are orthogonal if and only if the adjacent sides have equal magnitudes. This geometric relationship is mathematically proven by the dot product $(v+w) \cdot (v-w)$, which becomes zero the moment the lengths of vectors $\vec{v}$ and $\vec{w}$ align. Visual demonstrations highlight this by showing a 90-degree intersection and a red highlight only when these magnitudes are identical, effectively identifying the resulting shape as a rhombus. Ultimately, the rhombus is defined as the unique parallelogram where all four sides are equal, providing the necessary condition for its diagonals to be perpendicular.

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