📢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.