Example Data Table
| Vector A | Vector B | A × B | |A × B| | Area (triangle) |
|---|---|---|---|---|
| (1, 0, 0) | (0, 1, 0) | (0, 0, 1) | 1 | 0.5 |
| (0, 2, 0) | (3, 0, 0) | (0, 0, -6) | 6 | 3 |
| (2, -1, 4) | (-3, 5, 2) | (-22, -16, 7) | 27.4591 | 13.7296 |
Formula Used
For vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz):
|A × B| = √(Cx² + Cy² + Cz²)
Area(parallelogram) = |A × B|
Area(triangle) = |A × B| / 2
A · B = Ax·Bx + Ay·By + Az·Bz
cos(θ) = (A · B) / (|A||B|) when |A| and |B| are nonzero
How to Use
- Choose 3D for full vectors, or 2D for planar inputs.
- Enter components for Vector A and Vector B.
- Select decimals and your preferred output style.
- Click Calculate to view results above the form.
- Use CSV or PDF buttons to export the same results.
If either vector is zero, the angle is undefined.
Vector geometry and direction
The cross product converts two 3D vectors into a new vector that is perpendicular to both inputs. Its direction follows the right hand rule, so swapping the inputs flips the sign. This makes the operation useful for detecting orientation in coordinate systems, building surface normals in graphics, and describing rotational direction in mechanics.
Magnitude as scaled area
The magnitude of A × B equals |A||B|sin(θ), where θ is the angle between A and B. That value is the area of the parallelogram spanned by the vectors, and half of it is the area of the triangle they form. When vectors are parallel, sin(θ)=0 and the area collapses to zero. When vectors are perpendicular, sin(θ)=1 and the cross magnitude reaches its maximum for the given lengths, which is a helpful sanity check.
Angle and consistency checks
This calculator also reports the dot product and the angle when both magnitudes are nonzero. Together, dot and cross provide strong validation: dot describes alignment through cos(θ), while cross describes separation through sin(θ). If your vectors are nearly parallel, expect a small cross magnitude and an angle close to 0° or 180°. If either magnitude is zero, the angle is undefined, but the cross still evaluates correctly as the zero vector.
Unit normal for modeling
A unit normal is the normalized cross product, (A × B)/|A × B|, and it captures direction without scale. In modeling and CAD workflows, unit normals define the facing of a surface and influence lighting, backface culling, and collision responses. In engineering, the unit normal can represent a plane’s orientation for projecting forces or computing moment arms. When the cross magnitude is zero, a unique normal does not exist, so you must choose an alternate reference direction.
Practical input guidance
For stable results, keep units consistent and avoid mixing coordinate frames. In 2D mode, the calculator sets z to zero and returns a z-only cross component that represents signed area, which is useful for polygon orientation tests. Increase decimals when working with small angles or very large values, since rounding can hide meaningful differences. Export CSV or PDF to document calculations in reports, audits, and peer reviews, and include the input vectors alongside the computed vector.
FAQs
1) What does the cross product represent?
It returns a vector perpendicular to both inputs. Its direction follows the right hand rule, and its magnitude equals the parallelogram area spanned by the vectors.
2) Why is my result negative?
Sign depends on input order. A × B equals the negative of B × A. Swap vectors and the perpendicular direction reverses.
3) When is the angle undefined?
If either vector has zero magnitude, the angle between vectors cannot be computed. The cross product still evaluates and will be the zero vector.
4) How does 2D mode work here?
2D mode sets both z components to zero. The computed cross product points along the z axis, and its z value equals the signed area scale.
5) What is a unit normal used for?
It keeps only direction: (A × B)/|A × B|. Unit normals drive surface orientation, lighting calculations, and planar projections.
6) Why do my exports differ from what I see?
Exports use the same inputs, but rounding depends on the selected decimal setting. Increase decimals for tighter comparisons, especially with small angles.