Complex Cross Product Calculator

Vector inputs

Accepted forms: 3, -2.5, i, -i, 2+3i, -7-0.5i

Decimals

Output

Show steps

Vector A

aₓ aᵧ a_z

Vector B

bₓ bᵧ b_z

Options & checks

Uses: (aᵧb_z − a_zbᵧ, a_zbₓ − aₓb_z, aₓbᵧ − aᵧbₓ).
Complex multiply: (a+bi)(c+di)=(ac−bd)+(ad+bc)i.
Magnitude: sqrt(|cₓ|²+|cᵧ|²+|c_z|²).

Quick tip

Use -i for negative imaginary one, and avoid spaces.

Example data table

Use these input pairs to test the calculator.

Examples

Vector A	Vector B	A × B (manual form)	\|A × B\|	Note
(2+3i, 1−i, 4)	(−i, 3, 2−2i)	(aᵧb_z − a_zbᵧ, a_zbₓ − aₓb_z, aₓbᵧ − aᵧbₓ)	Computed by tool	Matches defaults
(1, i, 0)	(0, 1, i)	(−1, −i, 1)	1.732…	Fast verification
(3, −2i, 5+i)	(1+i, 4, −i)	Use determinant expansion	Computed by tool	Mixed components

Formula used

For A=(aₓ,aᵧ,a_z) and B=(bₓ,bᵧ,b_z):

A × B = (aᵧb_z − a_zbᵧ, a_zbₓ − aₓb_z, aₓbᵧ − aᵧbₓ)

Each operation uses complex arithmetic, and the output supports rectangular and polar display.

How to use this calculator

Enter three components for vector A and vector B.
Use formats like 2+3i or -i.
Choose decimals and output style, then enable steps if needed.
Press Submit to show the result above.
Download CSV history or a PDF report for sharing.

Complex vector cross product overview

This calculator handles 3D vectors with complex entries. Each component may be written as a real value, an imaginary term, or a full a+bi form. The cross product returns a third vector that is orthogonal in the algebraic sense. It is computed by determinant expansion.

Input formats and parsing rules

Supported patterns include 3, -2.5, i, -i, 4i, 2+3i, and -7-0.5i. Spaces are ignored. The tool converts the text into real and imaginary parts. It then applies complex multiplication and subtraction for every term.

Computation data and precision controls

You can choose 0 to 12 decimals. Rectangular output shows a+bi. Polar output shows r ∠ θ°. The magnitude uses |cₓ|, |cᵧ|, and |c_z|. It reports sqrt(|cₓ|²+|cᵧ|²+|c_z|²). This is a real scalar.

Graph interpretation in three modes

The 3D plot supports three views. Real parts show geometric trends in ℝ³. Imaginary parts isolate phase-only structure. Magnitudes show the size of each complex component. Vectors A, B, and A×B start at the origin. The axis range scales automatically.

Practical use cases and workflow

Complex vectors appear in phasor models, signal processing, and electromagnetic analysis. The cross product helps compute rotational effects and perpendicular directions. Use “Show steps” for audit trails. Export PDF for reports. Export CSV for batch checking in spreadsheets.

Session history and verification habits

The CSV file stores up to 50 recent runs in the current session. This makes quick comparisons easy. Validate results by swapping inputs. A×B should equal −(B×A). Also test simple vectors like (1,i,0) and (0,1,i) to confirm sign and order.

FAQs

1) What does “complex cross product” mean here?

It is the standard 3D cross product formula. The only change is that each component uses complex arithmetic. The output is a complex 3D vector.

2) Which input formats are accepted?

Use real numbers, pure imaginary terms, or a+bi forms. Examples include 3, -2.5, i, -i, 4i, 2+3i, and -7-0.5i. Avoid extra symbols.

3) Is the reported magnitude always real?

Yes. The tool computes magnitudes of complex components first, then combines them with a Euclidean norm. The final magnitude is a real, non‑negative scalar.

4) How should I interpret the unit direction output?

It scales each complex component by the real magnitude. This gives a normalized vector in a practical sense. If the cross product is zero, the unit direction is undefined.

5) What is the difference between rectangular and polar output?

Rectangular shows a+bi with signed real and imaginary parts. Polar shows r ∠ θ° using magnitude and angle in degrees. Both represent the same complex value.

6) What do the plot modes show?

Real parts plot the real components only. Imaginary parts plot the imaginary components only. Magnitudes plot |a|, |b|, and |c| for each axis component.