The generalized vector $S$, defined by contracting the Levi-Civita symbol $\varepsilon$ with the components of $N-1$ vectors $v_i$, is fundamentally guaranteed to be orthogonal to every single input vector $v_k$. This orthogonality arises because the dot product $S \cdot v_k$ necessarily introduces a repeated vector $\left(v_k\right)$ into the overall expression, which, due to the complete antisymmetry of the Levi-Civita symbol, forces the entire sum to vanish; this result is mathematically equivalent to the property that the determinant of a matrix with two identical columns (or rows) must be zero, establishing $S$ as the $N$-dimensional analog of the cross product.
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title The Geometry of Flux: A Roadmap to N-Dimensional Orthogonality
Resulmation: three-dimensional visualization of the cross product and the property of orthogonality
: N-Dimensional Normal Vector Demo and Electromagnetic Dual Visualization
IllustraDemo: Cross Product Guarantees Perfect Vector Perpendicularity
Ex-Demo: The Geometry of Orthogonal Engines and Universal Flux
Narr-graphic: N-Dimensional Orthogonality From Geometric Intuition to Tensor Calculus
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%% Proof and Derivation
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