The standard vector cross product is not defined for two vectors in two dimensions (often referred to informally as "two by two vectors" or 2D vectors).
When discussing vectors, the term "two by two vector" typically refers to a vector in a 2D space, having two components (e.g., [x, y]). The standard cross product operation is specifically defined for vectors in three-dimensional space (R³).
Understanding the Cross Product Definition
As the definition states, the cross product of two vectors results in "the third vector that is perpendicular to the two original vectors." Its magnitude relates to the area of the parallelogram formed by the vectors, and its direction is found using the right-hand rule.
Let's consider two vectors, A and B. In 3D space (R³), if A and B are not parallel, they define a plane. The cross product, A × B, is a vector that points directly away from or towards this plane, making it perpendicular to both A and B.
Why It Doesn't Work in Two Dimensions
The fundamental reason the standard cross product doesn't apply to two 2D vectors (resulting in a 2D vector) is the requirement for the resulting vector to be perpendicular to the original two.
- In a 2D plane, there is no unique "third" direction within that same plane that is simultaneously perpendicular to two non-parallel vectors.
- The direction perpendicular to a 2D plane exists in a higher dimension (the third dimension).
The Scalar "Cross Product" in 2D
While the vector cross product isn't defined in 2D space to produce a 2D vector, a related concept often arises, particularly in physics and engineering. This is often treated as a scalar value.
This scalar value is equivalent to the magnitude of the cross product if the two 2D vectors are considered as 3D vectors lying on the XY-plane (i.e., their z-component is zero).
Let A = [Ax, Ay] and B = [Bx, By].
Treat them as 3D vectors: A = [Ax, Ay, 0] and B = [Bx, By, 0].
The 3D cross product A × B would be:
A × B = [(Ay 0 - 0 By), (0 Bx - Ax 0), (Ax By - Ay Bx)]
A × B = [0, 0, (Ax By - Ay Bx)]
The result is a vector purely in the z-direction. The magnitude of this vector is |Ax By - Ay Bx|. This scalar value is what is often called the "cross product" of two 2D vectors.
- Magnitude: |Ax By - Ay Bx| represents the area of the parallelogram formed by the vectors A and B. This aligns with the magnitude definition from the reference.
- Sign: The sign of (Ax By - Ay Bx) indicates the orientation (clockwise or counterclockwise) of B relative to A. A positive value typically means B is counterclockwise from A, corresponding to the cross product pointing in the positive z-direction (out of the 2D plane).
This scalar value is also equivalent to the determinant of the matrix formed by the two vectors as rows or columns:
| Ax Ay |
| Bx By |Determinant = (Ax By) - (Ay Bx)
Summary
In conclusion, the standard vector cross product, which yields a third perpendicular vector, is not defined for two vectors in two dimensions. The perpendicular vector result requires a minimum of three dimensions. However, a related scalar quantity, equivalent to the magnitude of the 3D cross product (assuming z=0), is commonly calculated for 2D vectors and represents the oriented area of the parallelogram they form.
