Calculator
Enter values separated by commas, semicolons, or new lines. Accepted forms include 3+4i, 2-i, -5, i, and -2.5e-3+4i.
Example Data Table
| Case | Vector A | Vector B | Standard Hermitian inner product | Principal angle | Observation |
|---|---|---|---|---|---|
| Example 1 | [1+2i, 3-i, -2+0.5i] | [2-i, -1+4i, 0.5+2i] | -7 + 1.75i | 71.28° | General non-orthogonal complex vectors. |
| Example 2 | [1, i] | [2+3i, 4-i] | 1 - i | 79.48° | Short vectors with mixed real and imaginary parts. |
| Example 3 | [2-2i, -1+3i, 4] | [1+i, 2-2i, -3+0.5i] | -20 + 2i | 38.22° | Large magnitude overlap between components. |
Formula Used
Standard Hermitian inner product
⟨u, v⟩ = Σ conj(ui)vi
Alternative option
This page can also conjugate the second vector, or apply no conjugation, for comparison studies.
Vector norm
||u|| = √Σ |ui|2
Principal angle
θ = arccos(|⟨u, v⟩| / (||u|| ||v||))
Projection coefficient
c = ⟨u, v⟩ / ⟨u, u⟩, then proju(v) = cu
Norms and projection outputs on this page use the standard Hermitian form. That keeps geometric interpretations stable and mathematically consistent.
How to Use This Calculator
- Enter vector A values using commas or new lines.
- Enter vector B with the same number of components.
- Choose the conjugation rule that matches your convention.
- Set decimal precision for displayed results.
- Set a small tolerance for orthogonality checks.
- Press Calculate Inner Product.
- Review the result cards, step table, and Plotly graph.
- Use CSV or PDF download buttons for reporting.
FAQs
1. What is a complex vector inner product?
It is a scalar created by multiplying matching components and summing them. In complex spaces, one vector is usually conjugated first. That makes the result compatible with geometric length and angle concepts.
2. Why do we conjugate one vector?
Conjugation makes the inner product positive on nonzero self-products. Without it, lengths and angles can behave poorly. The standard Hermitian version uses conjugation on the first vector.
3. Which vector should be conjugated?
Many mathematics texts conjugate the first vector. Some engineering sources conjugate the second instead. This calculator lets you compare both conventions and also inspect the no-conjugation bilinear form.
4. What input formats are accepted?
You can enter values like 3+4i, 2-i, -5, i, and -i. Scientific notation is also accepted, such as 1e-3+2i. Separate entries with commas, semicolons, or line breaks.
5. What does orthogonal mean here?
Two vectors are orthogonal when the chosen inner product is zero. Because numerical rounding can create tiny residual values, the calculator checks against a configurable tolerance instead of exact zero.
6. Why can the principal angle show N/A?
The principal angle is displayed only for conjugate-based inner product rules. It is hidden when no conjugation is selected because the usual geometric angle formula may no longer be mathematically reliable.
7. What does the projection coefficient represent?
It tells how much of vector B lies along vector A. Multiply that coefficient by vector A to get the standard Hermitian projection of B onto A.
8. Can I export the calculation?
Yes. Use the CSV button for spreadsheet-friendly output. Use the PDF button for a clean report that includes summary metrics and the full component-by-component calculation table.