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

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.

### :clapper:three-dimensional visualization of the cross product and the property of orthogonality

> 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 $$v\_3$$.

{% embed url="<https://youtu.be/0au0yq209F4>" %}

### :pen\_ballpoint:Mathematical Proof

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