Calculator
Use comma, space, or line-separated values. Each vector must match the chosen dimension.
Formula Used
For vectors a = (a1, a2, ..., n) and b = (b1, b2, ..., bn),
the wedge product is
a∧b = Σ(i<j) (aᵢbⱼ - aⱼbᵢ) eᵢ∧eⱼ.
Each coefficient measures the signed oriented area projected onto a basis plane.
The overall magnitude is
||a∧b|| = √Σ(i<j)(aᵢbⱼ - aⱼbᵢ)²,
which equals the parallelogram area spanned by the two vectors.
In 2D, the result acts like one oriented scalar. In 3D, the coefficients correspond to the antisymmetric form related to the cross product.
How to Use This Calculator
- Select the vector dimension from 2D to 6D.
- Enter the components of Vector A and Vector B.
- Use commas, spaces, or line breaks between values.
- Set the decimal precision you want in the output.
- Click Calculate Wedge Product to generate the result.
- Review the bivector expression, coefficients, matrix, areas, and chart.
- Download the computed summary as CSV or PDF if needed.
Example Data Table
| Dimension | Vector A | Vector B | Main Wedge Output | Parallelogram Area |
|---|---|---|---|---|
| 2D | [3, 1] | [2, 4] | 10 e1∧e2 | 10 |
| 3D | [1, 2, 3] | [4, -1, 2] | -9 e1∧e2 - 10 e1∧e3 + 7 e2∧e3 | 15.1658 |
| 4D | [1, 0, 2, 1] | [3, 1, -2, 4] | 1 e1∧e2 - 8 e1∧e3 + 1 e1∧e4 - 2 e2∧e3 - 1 e2∧e4 + 10 e3∧e4 | 13.0767 |
FAQs
1) What does the wedge product represent?
It represents an oriented area element formed by two vectors. Instead of collapsing information into one length, it preserves directional plane information through signed basis-pair coefficients.
2) Why does the result contain several coefficients?
In dimensions above two, the wedge product lives in a bivector space. Each coefficient corresponds to one basis plane, such as e1∧e3, showing how strongly the vectors span that plane.
3) How is the area obtained from the wedge product?
Square every wedge coefficient, add them, and take the square root. That magnitude equals the parallelogram area. Divide by two if you need the triangle area.
4) What does a zero wedge product mean?
A zero result means the two vectors are linearly dependent, or nearly so within rounding tolerance. Geometrically, they fail to span an area because one vector points along the other.
5) How is this different from the dot product?
The dot product measures alignment and returns one scalar. The wedge product measures oriented spanning area and returns a bivector, which keeps plane-specific geometric information.
6) Is the wedge product the same as the cross product?
Not exactly. In 3D they are closely related, but the cross product is a pseudovector shortcut. The wedge product is more general and works naturally in any dimension.
7) What input format should I use?
Use commas, spaces, or separate lines. The only rule is that both vectors must contain exactly the same number of components as the selected dimension.
8) Why do signs change when I swap the vectors?
The wedge product is antisymmetric, so a∧b = -(b∧a). Swapping the vectors flips orientation, which reverses every bivector coefficient sign.