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.

🎬Narrated Video

https://youtu.be/RYLlOIxUVmY

🏗️Structural clarification of Poof and Derivation

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