# Tensors Define 3D Vector Direction

> The cross product is a mathematical operation unique to three-dimensional space that generates a vector perpendicular to two original vectors, with its direction governed by a convention known as the right-hand rule. This operation can be defined algebraically using the Levi-Civita tensor ($$\varepsilon\_{ijk}$$), a symbol whose value is determined by the order of its indices: cyclic permutations (such as 123 or 231) result in a positive value, anti-cyclic permutations (such as 132) result in a negative value, and any repeated indices result in zero. This tensor allows for a concise representation of the cross product through the formula $$\vec{v} \times \vec{w} = \vec{e}i \varepsilon{i j k} v^j w^k$$, which encapsulates fundamental unit vector identities where crossing two distinct unit vectors yields the third (e.g., $$vec{e}\_1 \times \vec{e}\_2 = \vec{e}\_3$$), while crossing a vector with itself always results in zero.

### :clapper:Narrated Video

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### :thread:Related Derivation

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[proving-the-cross-product-rules-with-the-levi-civita-symbol-cpr-lcs](https://via-dean.gitbook.io/all/~/revisions/WYsapCOkHSgcDPhymfq8/multifaceted-viewpoint/mathematical-structures-underlying-physical-laws/proof-and-derivation/proving-the-cross-product-rules-with-the-levi-civita-symbol-cpr-lcs)
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### :hammer\_pick:Compound Page

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