Case 1
- A — x
- 1
- A — y
- 2
- A — z
- 3
- B — x
- 4
- B — y
- 5
- B — z
- 6
7.348469
Open with these valuesCross Product Magnitude Calculator
7.348469
Result: 7.348469The cross product of (1, 2, 3) and (4, 5, 6) is (−3, 6, −3), and its length is √54 ≈ 7.3485. That length is the area of the parallelogram the two vectors span, so it is zero for parallel vectors and largest when they meet at a right angle.
The numbers at a glance
Held fixed: A — x 1.0000, A — y 2.0000, A — z 3.0000, B — x 4.0000, B — y 5.0000.
| B — z | Result |
|---|---|
| 0.0000 | 19.442222 |
| 2.0000 | 15.165751 |
| 4.0000 | 11.045361 |
| 6.0000Your value | 7.348469 |
| 8.0000 | 5.099020 |
| 10.0000 | 6.164414 |
| 12.0000 | 9.486833 |
Worked examples
Case 2
- A — x
- 1
- A — y
- 0
- A — z
- 0
- B — x
- 0
- B — y
- 1
- B — z
- 0
1.000000
Open with these valuesCase 3
- A — x
- 3
- A — y
- -3
- A — z
- 1
- B — x
- 4
- B — y
- 9
- B — z
- 2
41.833001
Open with these valuesHow it's calculated
|A × B| = √(cx² + cy² + cz²)
- StepEnter the three components of A, then those of B.
- StepThe cross product is cx = ay·bz − az·by, cy = az·bx − ax·bz, cz = ax·by − ay·bx.
- ResultIts length √(cx² + cy² + cz²) is the area of the parallelogram.
Reference table
| ax, ay, az, bx, by, bz | A × B | Magnitude |
|---|---|---|
| 1, 0, 0, 2, 0, 0 | (0, 0, 0) | 0.000000 |
| 1, 0, 0, 0, 1, 0 | (0, 0, 1) | 1.000000 |
| 2, 0, 0, 0, 3, 0 | (0, 0, 6) | 6.000000 |
| 1, 2, 3, 4, 5, 6 | (−3, 6, −3) | 7.348469 |
| -1, -2, -3, -4, -5, -6 | (−3, 6, −3) | 7.348469 |
| 3, -3, 1, 4, 9, 2 | (−15, −2, 39) | 41.833001 |
Questions
How do I calculate the magnitude of a cross product?
First find the cross product A × B = (ay·bz − az·by, az·bx − ax·bz, ax·by − ay·bx), then take its length √(cx² + cy² + cz²). For (1, 2, 3) and (4, 5, 6) the cross product is (−3, 6, −3), so the magnitude is √54 ≈ 7.3485.
What does the cross product magnitude represent?
It equals the area of the parallelogram spanned by the two vectors. Geometrically it is |A| |B| sin θ, where θ is the angle between them, so it grows as the vectors become more perpendicular.
When is the cross product magnitude zero?
Whenever the two vectors are parallel or anti-parallel, because sin θ is then zero. It is also zero if either vector is the zero vector, and it is largest when the vectors meet at a right angle.
How is the cross product different from the dot product?
The dot product returns a single number and uses cos θ, so it is largest for parallel vectors. The cross product returns a vector perpendicular to both inputs and uses sin θ, so its length is largest for perpendicular vectors.
Does the cross product work for 2D vectors?
The cross product is defined for 3D vectors, so treat a 2D vector as (x, y, 0). The result then points along the z axis and its length is the area of the parallelogram in the plane.
Sources and last check
Information, not professional advice.