Cross Product Magnitude
Enter the components of two 3D vectors to get the magnitude of their cross product — the length of A × B, which equals the area of the parallelogram the two vectors span.
Magnitude of A × B at a glance
Enter the x, y, and z components of vectors A and B and this cross product calculator returns |A × B| = √(cx² + cy² + cz²) instantly.
Order matters
A × B and B × A point in opposite directions but share the same magnitude, so the length you get back does not depend on the order you enter them.
What is the cross product magnitude?
The length of A × B
This cross product calculator works out the magnitude — the length — of the cross product of two 3D vectors. The cross product A × B is itself a vector that points perpendicular to both A and B, and its length tells you how much "turning" the two vectors produce together. That length equals the area of the parallelogram the two vectors span, which is why it shows up everywhere from torque and angular momentum in physics to surface normals and lighting in 3D graphics. Enter the three components of each vector and the calculator returns the single number |A × B|.
Enter the components of two 3D vectors to get the magnitude of their cross product — the area of the parallelogram they span — instantly.
The cross product first builds a new vector from the components, then the magnitude is the length of that vector.
|A × B| = √(cx² + cy² + cz²)The three components of A × B are cx = ay·bz − az·by, cy = az·bx − ax·bz, and cz = ax·by − ay·bx. Once you have those, the magnitude is just the square root of the sum of their squares — exactly the same length formula used for any 3D vector.
Suppose A = (1, 2, 3) and B = (4, 5, 6).
Build the cross product vector
cx = 2·6 − 3·5 = −3, cy = 3·4 − 1·6 = 6, cz = 1·5 − 2·4 = −3, so A × B = (−3, 6, −3).
Square and sum the components
(−3)² + 6² + (−3)² = 9 + 36 + 9 = 54.
Take the square root
√54 ≈ 7.3485 — the magnitude of A × B, and the area of the parallelogram the two vectors span.
The magnitude has a clean geometric meaning: |A × B| = |A| |B| sin θ, where θ is the angle between the two vectors. Because of the sin θ factor, the result is largest when the vectors are perpendicular (sin 90° = 1, giving |A| |B|) and shrinks to zero as they line up. That same quantity is the area of the parallelogram with A and B as its sides, so a larger magnitude means the two vectors "open up" more between them. This is why the cross product magnitude is the natural tool for measuring torque (force times lever arm), the strength of a rotational effect, or the area of a triangle (half the cross product magnitude of two of its edges).
The formula is exact, but a few properties of the cross product are worth keeping in mind.
Three dimensions, and zero for parallel vectors
The cross product is defined for 3D vectors — use a z component of 0 to handle a 2D vector. The magnitude is zero whenever the two vectors are parallel or anti-parallel (or one is the zero vector), because |A × B| = |A| |B| sin θ and sin θ = 0 then. The magnitude alone discards direction: A × B and B × A return the same length even though the cross product vectors point opposite ways.