Calculator
Example Data Table
| Vector A | Vector B | A × B | |A × B| |
|---|---|---|---|
| (2, 3, 4) | (1, 5, 2) | (-14, 0, 7) | 15.6525 |
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | 1 |
| (3, -1, 2) | (4, 2, -5) | (1, 23, 10) | 25.0998 |
Formula Used
The cross product of two 3D vectors A = (ax, ay, az) and B = (bx, by, bz) is:
A × B = (aybz - azby, azbx - axbz, axby - aybx)
The magnitude of the result is:
|A × B| = √(cx² + cy² + cz²)
This magnitude equals the area of the parallelogram formed by the two vectors.
How to Use This Calculator
- Enter the three components of Vector A.
- Enter the three components of Vector B.
- Select row or column matrix output.
- Choose the decimal precision you want.
- Enable steps, unit vector, or validation if needed.
- Click the calculate button.
- Review the result shown above the form.
- Download CSV or PDF when required.
Matrix Cross Product Guide
What this calculator does
This matrix cross product calculator finds the cross product of two three dimensional vectors. It accepts matrix style inputs and returns a new vector that is perpendicular to both originals. The tool is useful for maths study, mechanics, graphics, geometry, and engineering work.
Why the cross product matters
The cross product helps measure direction and area at the same time. Its output follows the right hand rule. That makes it important when you need a normal vector for a plane, a turning direction in space, or the area of a parallelogram formed by two vectors.
Matrix form and vector meaning
Students often write vectors as row matrices or column matrices. This calculator supports both display styles. The underlying calculation remains the same. Only the presentation changes. That makes the tool suitable for classroom notation, homework format, and exam revision.
Built for checking accuracy
This calculator does more than show one answer. It also reports vector magnitudes, angle between inputs, area, and dot product checks. These checks help confirm that the resulting vector is orthogonal to both starting vectors. When the result becomes zero, the vectors are parallel or one vector is zero.
Helpful for many subjects
In coordinate geometry, the cross product helps find plane normals. In physics, it appears in torque, angular momentum, and magnetic force problems. In computer graphics, it supports lighting, surface orientation, and camera movement. In engineering, it helps with spatial modelling and direction analysis.
Fast export and clean workflow
The calculator includes export options for quick reporting. Use CSV for spreadsheets and result logs. Use PDF for assignments or printable summaries. Because the form sits below the result area, you can review your output instantly after submission and then adjust values without leaving the page.
FAQs
1. What is a matrix cross product?
It is the cross product of two 3D vectors written in matrix form. The result is another vector perpendicular to both inputs.
2. Can I use row matrices and column matrices?
Yes. This calculator lets you display the answer as either a row matrix or a column matrix. The numeric result stays the same.
3. Why is the result perpendicular to both vectors?
That is a defining property of the cross product. The validation section checks this by computing dot products with each original vector.
4. What does a zero cross product mean?
A zero result means the vectors are parallel, anti-parallel, or one vector has zero magnitude. In that case, no unique perpendicular direction exists.
5. What is the meaning of |A × B|?
It is the magnitude of the cross product. It also equals the area of the parallelogram formed by the two input vectors.
6. Does this work for 2D vectors?
This version is designed for three dimensional vectors. A 2D problem can be embedded into 3D by using zero as the third component.
7. Why does the calculator show determinant steps?
They help you verify the x, y, and z components manually. This is useful for learning, checking classwork, and reducing algebra mistakes.
8. When should I download CSV or PDF?
Use CSV for spreadsheet records and repeated analysis. Use PDF when you need a neat file for printing, sharing, or submitting coursework.