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

$\complement\cdots$Counselor

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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

Surface Parametrisation and the Verification of the Gradient-Normal Relationship

Proof and Implications of a Vector Operator Identity

Conditions for a Scalar Field Identity

Solution and Proof for a Vector Identity and Divergence Problem

Kinematics and Vector Calculus of a Rotating Rigid Body

Work Done by a Non-Conservative Force and Conservative Force

The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field

Calculating the Area of a Half-Sphere Using Cylindrical Coordinates

Divergence Theorem Analysis of a Vector Field with Power-Law Components

Total Mass in a Cube vs. a Sphere

Momentum of a Divergence-Free Fluid in a Cubic Domain

Total Mass Flux Through Cylindrical Surfaces

Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field

Computing the Integral of a Static Electromagnetic Field

Surface Integral to Volume Integral Conversion Using the Divergence Theorem

Circulation Integral vs. Surface Integral

Using Stokes' Theorem with a Constant Scalar Field

Verification of the Divergence Theorem for a Rotating Fluid Flow

Integral of a Curl-Free Vector Field

Boundary-Driven Cancellation in Vector Field Integrals

The Vanishing Curl Integral

Proving the Generalized Curl Theorem

Computing the Magnetic Field and its Curl from a Dipole Vector Potential

Proving Contravariant Vector Components Using the Dual Basis

Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates

Vector Field Analysis in Cylindrical Coordinates

Vector Field Singularities and Stokes' Theorem

Compute Parabolic coordinates-related properties

Analyze Flux and Laplacian of The Yukawa Potential

Verification of Vector Calculus Identities in Different Coordinate Systems

Analysis of a Divergence-Free Vector Field

The Uniqueness Theorem for Vector Fields

Analysis of Electric Dipole Force Field

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