📢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.

🎬Narrated Video

Description

This illustration, titled "Visualizing the 3D Vector Cross Product," provides a visual and mathematical guide to how the cross product functions, specifically using the Levi-Civita Symbol ( $\epsilon_{ijk}$ ).

The illustration is divided into two primary sections:

1. The Cross Product: Rules & Properties

This section explains the geometric behavior of vectors during a cross product:

Right-Hand Rule: It illustrates that the cross product of two vectors results in a third vector ( $e_3$ ) that is orthogonal (perpendicular) to both original vectors ( $e_1$ and $e_2$ ).
Parallel Vectors: It explicitly shows that the cross product of any vector with itself (parallel vectors) is always zero (e.g., $e_1 \times e_1 = 0$ ).
Basis Vector Cycle: A circular diagram demonstrates the cyclic nature of orthogonal basis vectors:
$e_1 \times e_2 = e_3$
$e_2 \times e_3 = e_1$
$e_3 \times e_1 = e_2$

2. The Levi-Civita Symbol

This section introduces the symbol as a "compact formula" that simplifies cross-product calculations into a single equation. The symbol's value is determined by the permutation of its indices ( $i, j, k$ ):

Even (Cyclic) Permutations: When indices follow the order 123, 231, or 312, the value is +1.
Odd (Anti-Cyclic) Permutations: When indices follow the order 132, 321, or 213, the value is 1.
Repeated Indices: While not explicitly detailed in the value timeline, the symbol accounts for cases where the result is 0.