🧄A parallelogram is a rhombus (has equal sides) if and only if its diagonals are perpendicular (PRD)

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

🎬Parallelogram Diagonals Orthogonality Demo

📎IllustraDemo

📢Diagonals Are Perpendicular Only In A Rhombus

🧣Example-to-Demo

🧣Vector Proofs of Rhombus Orthogonality (VP-RO)

🍁Vector Geometry of Quadrilaterals proves the Rhombus Condition

Description

These three documents collectively illustrate a geometric proof using vector algebra to demonstrate that a parallelogram is a rhombus if and only if its diagonals are perpendicular. By defining the sides of a parallelogram as vectors $\vec{v}$ and $\vec{w}$, the diagonals are represented as their sum $(\vec{v} + \vec{w})$ and difference $(\vec{v} - \vec{w})$. The mathematical core of the argument relies on the dot product of these diagonals; when set to zero to represent orthogonality, the expression simplifies to $\|\vec{v}\|^2 - \|\vec{w}\|^2 = 0$, proving that the side lengths must be equal ($\|\vec{v}\| = \|\vec{w}\|$). This logic allows for a clear classification of geometric states, ranging from a general parallelogram with unequal sides to a square, which requires both equal sides and orthogonal base vectors.

Key Takeaways from the Visuals

• The Vector Setup: Adjacent sides of a parallelogram are modeled as vectors $\vec{v}$ and $\vec{w}$, while the diagonals are defined by the vector operations $(\vec{v} + \vec{w})$ and $(\vec{v} - \vec{w})$.

• The Orthogonality Test: For diagonals to cross at a $90^\circ$ angle, their dot product must equal zero: $(\vec{v} + \vec{w}) \cdot (\vec{v} - \vec{w}) = 0$.

• Algebraic Simplification: Expanding the dot product using the distributive property and commutativity results in the identity $\|\vec{v}\|^2 - \|\vec{w}\|^2 = 0$.

• Geometric Classification:

  • Parallelogram: $\|\vec{v}\| \neq \|\vec{w}\|$; diagonals are not orthogonal.

  • Rhombus: $\|\vec{v}\| = \|\vec{w}\|$; diagonals are orthogonal.

  • Square: $\|\vec{v}\| = \|\vec{w}\|$ and $\vec{v} \cdot \vec{w} = 0$ (sides are perpendicular).

• Application & Demonstration: The concepts can be validated through Python computations for plotting vectors or interactive HTML tools to explore the relationship between side magnitudes and diagonal angles.

⚒️Compound Page

A parallelogram is a rhombus (has equal sides) if and only if its diagonals are perpendicular | Provationviadean on Notion