Cross Product Calculator
A cross product calculator finds A x B for two 3D vectors: the vector perpendicular to both inputs, with step-by-step component working. You also get the magnitude, the parallelogram area, the unit normal, and B x A to see the sign flip. New: a 3D visualization draws your vectors, the result, and the parallelogram they span.
The visualization is a perspective-style projection of your actual inputs, not a generic stock diagram.
What the cross product is
The cross product A x B of two 3D vectors is a third vector that is perpendicular to both of them, with its direction given by the right-hand rule: point your fingers along A, curl toward B, and your thumb points along A x B. Its magnitude equals the area of the parallelogram the two vectors span, which is |A||B|sin(theta). Component by component: A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx).
How to use this calculator
Enter the x, y and z components of vectors A and B, then calculate. The tool shows each component's working, the resulting vector, its magnitude, the parallelogram area (the magnitude itself), the triangle area (half), the unit normal direction, and B x A to demonstrate the sign flip. If the result is zero, the tool tells you the vectors are parallel. The 3D diagram below the numbers draws your actual vectors as arrows, the result in red, and the shaded parallelogram.
Worked example
A = (1, 2, 3), B = (4, 5, 6): x: 2 x 6 - 3 x 5 = 12 - 15 = -3; y: 3 x 4 - 1 x 6 = 12 - 6 = 6; z: 1 x 5 - 2 x 4 = 5 - 8 = -3. So A x B = (-3, 6, -3), and |A x B| = sqrt(9 + 36 + 9) = sqrt(54) = 7.348469. Tool output matches exactly. The visualization shows A in blue, B in amber, the result in red pointing perpendicular to both, and the shaded parallelogram between A and B whose area is 7.348469.
New: the 3D visualization
Competitor tools with interactive 3D views help users see what the cross product means geometrically, so this page now draws one from your own inputs with no plugins or libraries. The diagram projects your vectors into a 3D-style view: the x, y, and z axes in grey, A in blue, B in amber, and A x B in red shooting off perpendicular to both. The shaded parallelogram between A and B makes the magnitude's meaning concrete: its area is |A x B|. For parallel inputs the diagram collapses to a line, which is exactly why the result is the zero vector.
What changed in this version
The 3D visualization is the only addition. All the numbers (component steps, magnitude, parallelogram and triangle areas, unit normal, B x A) are computed by the same code as before, unchanged.
Frequently asked questions
What is the cross product used for?
Finding a vector perpendicular to two given vectors (surface normals in graphics), computing areas of parallelograms and triangles in 3D, and torque in physics (r x F).
Why is B x A the negative of A x B?
The cross product is anti-commutative: swapping the inputs flips the direction the right-hand rule gives. The tool shows both so you can see the sign flip on your own numbers.
What does a zero cross product mean?
The vectors are parallel (or one is zero): the parallelogram they span has zero area, so there is no unique perpendicular direction. The unit normal is undefined in that case.
Is the diagram my actual vectors?
Yes. It projects the exact components you entered; it is not a generic stock diagram. Parallel inputs collapse the parallelogram to a line, exactly matching the math.