# A parallelogram is a rhombus (has equal sides) if and only if its diagonals are perpendicular > The vector analysis of the parallelogram diagonals, $$v+w$$ and $$v-w$$, yields a powerful geometric conclusion: their orthogonality is mathematically equivalent to the condition $$(v+ w) \cdot(v-w)=0$$. By expanding this dot product using the distributive property, the cross terms $$(v \cdot w)$$ cancel out due to the dot product's commutativity, leaving the simplified expression $$|v|^2-|w|^2$$. Therefore, for the diagonals to be orthogonal, this expression must equal zero, which directly implies that $$|v|^2=|w|^2$$, or simply $$|v|=|w|$$. The key takeaway is that in any parallelogram, the diagonals are perpendicular if and only if the adjacent vectors ( $$v$$ and $$w$$ ) have equal magnitudes, meaning the parallelogram is a rhombus. ### :clapper:Narrated Video {% content-ref url="../animated-results/parallelogram-diagonals-orthogonality-demo" %} [parallelogram-diagonals-orthogonality-demo](https://via-dean.gitbook.io/all/~/revisions/WYsapCOkHSgcDPhymfq8/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/animated-results/parallelogram-diagonals-orthogonality-demo) {% endcontent-ref %} ### :paperclip:IllustraDemo {% content-ref url="../illustrademo/diagonals-are-perpendicular-only-in-a-rhombus" %} [diagonals-are-perpendicular-only-in-a-rhombus](https://via-dean.gitbook.io/all/~/revisions/WYsapCOkHSgcDPhymfq8/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/illustrademo/diagonals-are-perpendicular-only-in-a-rhombus) {% endcontent-ref %} ### :scarf:Example-to-Demo {% content-ref url="../example-to-demo/vector-proofs-of-rhombus-orthogonality-vp-ro" %} [vector-proofs-of-rhombus-orthogonality-vp-ro](https://via-dean.gitbook.io/all/~/revisions/WYsapCOkHSgcDPhymfq8/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/example-to-demo/vector-proofs-of-rhombus-orthogonality-vp-ro) {% endcontent-ref %} ### :hammer\_pick:Compound Page {% embed url="" %}