Enter Complex Vector Values
Use real and imaginary fields for each component. The calculator supports 2D and 3D complex vectors and computes multiple advanced results in one submission.
Formula Used
Vector addition: A + B = [a1 + b1, a2 + b2, a3 + b3]
Vector subtraction: A - B = [a1 - b1, a2 - b2, a3 - b3]
Hermitian inner product: ⟨A,B⟩ = Σ conjugate(ak)bk
Bilinear dot product: A · B = Σ akbk
Norm: ||A|| = √Σ|ak|²
Unit vector: Â = A / ||A||, when ||A|| ≠ 0
Projection of A onto B: projB(A) = (⟨A,B⟩ / ⟨B,B⟩) B
Distance: ||A - B||
Principal angle: θ = cos-1(|⟨A,B⟩| / (||A|| ||B||))
3D cross product: A × B = [a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1]
How to Use This Calculator
- Choose whether your vectors are 2D or 3D.
- Set the output precision for cleaner formatting.
- Enter the real and imaginary parts for every component of Vector A.
- Enter the real and imaginary parts for every component of Vector B.
- Click the calculation button to generate all results at once.
- Review vector sums, products, norms, projections, angles, and distance.
- Use the chart to compare magnitudes by component.
- Download the generated results as CSV or PDF whenever needed.
Example Data Table
| Case | Dimension | Vector A | Vector B | A + B | ⟨A,B⟩ | ||A|| | ||A-B|| |
|---|---|---|---|---|---|---|---|
| 1 | 2D | [2 + 1i, -1 + 2i] | [1 - 2i, 4 + 1i] | [3 - 1i, 3 + 3i] | -2 - 14i | 3.162 | 6 |
| 2 | 3D | [3, 2i, -1 + 1i] | [1 + 1i, -2, 2 - 3i] | [4 + 1i, -2 + 2i, 1 - 2i] | -2 + 8i | 3.873 | 6.164 |
| 3 | 3D | [0.5 - 1.5i, 2 + 0.75i, 1] | [-1 + 2i, 3 - 0.5i, 2i] | [-0.5 + 0.5i, 5 + 0.25i, 1 + 2i] | 2.125 - 1.75i | 2.839 | 4.697 |
Frequently Asked Questions
1. What is a complex vector?
A complex vector is a vector whose components contain real and imaginary parts. Instead of ordinary real-number entries, each component may look like a + bi.
2. Why does this calculator use the Hermitian inner product?
For complex vectors, the Hermitian inner product is standard because it conjugates one vector before multiplication. That keeps norms real and nonnegative.
3. What is the difference between the Hermitian inner product and the bilinear dot product?
The Hermitian form uses conjugates and is common in complex vector spaces. The bilinear dot product multiplies components directly without conjugation and is included for comparison.
4. Can I use the calculator for 2D and 3D vectors?
Yes. Choose 2D for x and y components only, or choose 3D to include z values. The third component fields automatically appear when needed.
5. When is the cross product available?
The cross product is shown only for 3D vectors. In 2D mode, the calculator hides that result because the standard vector cross product requires three components.
6. Why might a unit vector or projection be undefined?
A unit vector needs a nonzero magnitude. A projection onto B also requires B to be nonzero. If either denominator becomes zero, that result is undefined.
7. What does the plotted graph represent?
The graph compares component magnitudes for Vector A, Vector B, and their sum. It helps you quickly inspect how strongly each component contributes.
8. Can I save the results for reports or coursework?
Yes. After calculation, use the CSV button for spreadsheet-friendly data or the PDF button for a clean report-style export of the main results.