Calculator Input
Enter 3D vector components. For 2D work, set both z-values to 0.
Formula Used
A unit vector orthogonal to two non-parallel vectors comes from the normalized cross product. The vector order sets the direction. Reversing the order flips the sign.
If A = (ax, ay, az) and B = (bx, by, bz), then:
A × B = ( aybz - azby, azbx - axbz, axby - aybx )
|A × B| = √(cx2 + cy2 + cz2)
n = (A × B) / |A × B|
This makes |n| = 1, while A · n = 0 and B · n = 0.
How to Use This Calculator
- Enter the x, y, and z components for Vector A.
- Enter the x, y, and z components for Vector B.
- Choose A × B or B × A for the preferred direction.
- Set your desired decimal precision.
- Click the calculate button.
- Read the unit vector, opposite unit vector, and verification values.
- Use the CSV or PDF button to save the result.
- Inspect the Plotly graph to compare vector directions visually.
Example Data Table
| Vector A | Vector B | Order | Cross Product | Unit Orthogonal Vector |
|---|---|---|---|---|
| (1, 0, 0) | (0, 1, 0) | A × B | (0, 0, 1) | (0.0000, 0.0000, 1.0000) |
| (2, 1, 0) | (1, 3, 0) | A × B | (0, 0, 5) | (0.0000, 0.0000, 1.0000) |
| (1, 2, 3) | (4, 5, 6) | A × B | (-3, 6, -3) | (-0.4082, 0.8165, -0.4082) |
| (3, -1, 2) | (1, 4, -2) | B × A | (6, -8, -13) | (0.3676, -0.4901, -0.7964) |
Frequently Asked Questions
1) What does this calculator find?
It finds a unit vector perpendicular to two input vectors. It also shows the raw cross product, vector magnitudes, dot-product checks, angle information, and the opposite unit direction.
2) Why does the cross product give an orthogonal vector?
In three dimensions, the cross product is constructed so its result is perpendicular to both input vectors. Dividing that result by its magnitude creates a unit-length orthogonal vector.
3) What happens if the vectors are parallel?
The cross product becomes the zero vector. A zero vector cannot be normalized, so no unique orthogonal unit vector is produced by this method.
4) Why are there two valid unit vectors?
If n is orthogonal and unit length, then -n is also orthogonal and unit length. Both are correct. They point in opposite directions along the same normal line.
5) Does vector order matter?
Yes. A × B and B × A have equal magnitudes but opposite directions. This calculator lets you choose the order so the normal matches your preferred orientation.
6) Can I use this for 2D vectors?
Yes. Enter z = 0 for both vectors. The result will typically point along the positive or negative z-axis, depending on the chosen vector order.
7) How can I verify the result quickly?
Check that A · n = 0, B · n = 0, and |n| = 1. Those three tests confirm perpendicularity to both vectors and correct unit length.
8) What does the Plotly graph show?
The graph shows Vector A, Vector B, the selected cross product, and the unit normal from the origin. It helps you compare direction, relative size, and orientation.