🧄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.

🎬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 $v_3$.

🖊️Mathematical Proof

The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation | Proof and Derivationviadean on Notion