The proof uses the property of the Levi-Civita symbol to show that the dot product of the cross-product vector $S$ and any of the original vectors $v_k$ is zero. This is because the index notation creates a repeated index, which makes the symbol (and thus the dot product) vanish. The animation visually confirms this principle, showing $S$ remaining perpendicular to the two vectors that created it, while its dot product with an arbitrary third vector is non-zero, proving the orthogonality is specific.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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The proof demonstrates that the vector $S$ is orthogonal to all original vectors $v_k$ by showing their dot product is zero. The argument hinges on a crucial property of the Levi-Civita symbol.

• Proof Method: The proof uses index notation to express the dot product $S \cdot v_k$. The core goal is to show this expression evaluates to zero.
• The Key Player: The Levi-Civita symbol ( $\varepsilon$ ) is the central element of the proof. Its fundamental property is that its value is zero if any two of its indices are the same. It's non-zero only when all its indices are unique.
• The Final Step: When the definition of $S$ is substituted into the dot product formula, the summation over indices forces one of the indices from the dot product term ( $a$ ) to become identical to one of the indices from the Levi-Civita symbol ( $i_k$ ). This creates a repeated index in the symbol, causing its value to become zero. Consequently, the entire dot product expression becomes zero, proving the orthogonality of $S$ to all $v_k$.

🎬Demonstration

The animated demo visualizes how the cross product of two vectors, $v_1$ and $v_2$, dynamically generates a third vector, $S$, that remains perpetually orthogonal to both. It further demonstrates that this orthogonality is specific to the two source vectors by showing that $S$ is not necessarily orthogonal to an arbitrary third vector, $v_3$. This is confirmed by live dot product calculations which are zero for $v_1$ and $v_2$ but non-zero for $v_3$.

three-dimensional visualization of the cross product and the property of orthogonality

three-dimensional visualization of the cross product and the property of orthogonality

✍️Mathematical Proof

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Proving the Cross Product Rules with the Levi-Civita Symbol

Proving the Epsilon-Delta Relation and the Bac-Cab Rule

Simplifying Levi-Civita and Kronecker Delta Identities

Dot Cross and Triple Products

Why a Cube's Diagonal Angle Never Changes

How the Cross Product Relates to the Sine of an Angle

Finding the Shortest Distance and Proving Orthogonality for Skew Lines

A Study of Helical Trajectories and Vector Dynamics

The Power of Cross Products: A Visual Guide to Precessing Vectors

Divergence and Curl Analysis of Vector Fields

Unpacking Vector Identities: How to Apply Divergence and Curl Rules

Commutativity and Anti-symmetry in Vector Calculus Identities

Double Curl Identity Proof using the epsilon-delta Relation

The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation

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