# 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.

### :paperclip:IllustraDemo

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### :thread:Related Derivation

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[a-parallelogram-is-a-rhombus-has-equal-sides-if-and-only-if-its-diagonals-are-perpendicular](https://via-dean.gitbook.io/all/~/revisions/WYsapCOkHSgcDPhymfq8/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/proof-and-derivation/a-parallelogram-is-a-rhombus-has-equal-sides-if-and-only-if-its-diagonals-are-perpendicular)
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### :hammer\_pick:Compound Page

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