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2D Vector Tensor Product Calculator — result sheet
Calculate the outer tensor product of two real two-dimensional vectors and report its four components, Frobenius norm, and rank-one consistency check.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: The outer tensor product a⊗b is the 2×2 matrix [[a₁b₁, a₁b₂],[a₂b₁, a₂b₂]]. Its Frobenius norm is ‖a⊗b‖F = ‖a‖₂‖b‖₂, and its 2×2 determinant is zero in exact arithmetic.
A tensor product is not the same operation as a dot product or ordinary matrix multiplication. This page uses two 2D vectors and builds every matrix entry explicitly, then reports the norm identity and determinant residual so the visitor can distinguish the outer-product structure from other familiar products.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- First vector component a₁ — minimum -1000000; maximum 1000000
- First vector component a₂ — minimum -1000000; maximum 1000000
- Second vector component b₁ — minimum -1000000; maximum 1000000
- Second vector component b₂ — minimum -1000000; maximum 1000000
Worked example
| Input | Value |
|---|---|
| First vector component a₁ | 2 |
| First vector component a₂ | -1 |
| Second vector component b₁ | 3 |
| Second vector component b₂ | 4 |
a⊗b = [[6,8],[−3,−4]], Frobenius norm = √130, and determinant = 0.
Assumptions and limits
- Both inputs are real two-dimensional vectors.
- The displayed 2×2 matrix uses the first vector as a column and the second as a row in the outer-product convention.
- The tensor product is not a dot product and does not reduce to one scalar.
- The determinant should be zero for exact arithmetic because the matrix has rank at most one.
- Floating-point multiplication may leave a tiny nonzero determinant residual.
- The page does not construct higher-order tensors, Kronecker products of matrices, or a basis transformation.
- Units, if the vector components carry them, follow the product of the two vector units.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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