Formula Used
For vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃), the cross product is:
A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
The magnitude is:
|A × B| = |A||B|sinθ
This magnitude also gives the area of the parallelogram formed by both vectors.
How to Use This Calculator
Enter the three components of vector A.
Enter the three components of vector B.
Select the operation order. Remember that A × B and B × A have opposite signs.
Use the scale factor when the final vector must be multiplied.
Press Calculate to show the result above the form.
Use CSV or PDF buttons to save the calculated output.
Example Data Table
| Vector A | Vector B | A × B | Magnitude |
|---|---|---|---|
| (3, -3, 1) | (4, 9, 2) | (-15, -2, 39) | 41.833 |
| (1, 2, 3) | (4, 5, 6) | (-3, 6, -3) | 7.3485 |
| (2, 0, 1) | (1, -1, 3) | (1, -5, -2) | 5.4772 |
Understanding the Cross Product
The cross product is a vector operation for three dimensional vectors. It creates a new vector that is perpendicular to both input vectors. This makes it useful in geometry, physics, mechanics, graphics, and engineering design.
Why Direction Matters
The direction follows the right hand rule. Curl your fingers from vector A toward vector B. Your thumb points along A × B. Reversing the order changes the sign of every component. This is why B × A is not the same as A × B.
Magnitude Meaning
The magnitude of the cross product shows the area of the parallelogram formed by both vectors. Half of that value gives the triangle area. When the vectors are parallel, the magnitude becomes zero. That means the vectors do not span any area.
3x3 Determinant View
The calculation can be written as a 3x3 determinant. The first row uses i, j, and k unit directions. The second row holds vector A. The third row holds vector B. Expanding that determinant gives the three output components.
Practical Uses
In physics, cross products describe torque, angular momentum, and magnetic force. In computer graphics, they create surface normals for lighting. In surveying and design, they help measure orientation and area. This calculator also gives angle, dot product, magnitude, and unit normal details.
Input Care
Use signed values carefully. Negative components change the final direction. Decimal values are accepted. The optional scale factor lets you multiply the final vector without changing the input values. The unit label is only a display label. It does not change the mathematics.
Result Review
After calculation, compare the magnitude with the angle. A small angle usually gives a smaller cross product. A ninety degree angle gives the largest magnitude for fixed vector lengths. Export the result when you need a record for homework, reports, or design checks.
FAQs
What is a 3x3 cross product calculator?
It calculates the vector product of two three dimensional vectors. The result is another vector perpendicular to both original vectors.
Why does the calculator show three result components?
A cross product in three dimensions has x, y, and z components. These components describe the final perpendicular vector.
Is A × B equal to B × A?
No. The magnitudes are equal, but the directions are opposite. Reversing the order changes the sign of each component.
What does the magnitude mean?
The magnitude equals the area of the parallelogram formed by the two vectors. Half of it gives the triangle area.
When is the cross product zero?
It is zero when the vectors are parallel, anti-parallel, or when either vector has zero length.
Can I use decimal values?
Yes. The calculator accepts whole numbers, negative numbers, and decimal values for all vector components.
What is the unit normal vector?
It is the cross product divided by its magnitude. It keeps direction but changes the length to one.
Why include CSV and PDF downloads?
They help save calculation records. CSV is useful for spreadsheets. PDF is useful for reports, notes, and submissions.