Cross Product Calculator

This calculator finds the cross product of two three-dimensional vectors, a new vector that points at right angles to both of them, along with its magnitude and the area of the parallelogram the two vectors span. The cross product is a fundamental operation in vector algebra, with no equivalent in two dimensions, and it answers a question that comes up constantly in physics, engineering and computer graphics: given two directions, what direction is perpendicular to both? The result is a vector whose direction follows the right-hand rule and whose length equals the area of the parallelogram formed by the two input vectors. This makes the cross product the natural tool for finding normals to surfaces in 3D graphics, calculating torque and angular momentum in physics, determining the moment of a force, and testing whether vectors are parallel, since the cross product of parallel vectors is zero. This tool computes it for you. You enter the x, y and z components of each vector, and the calculator returns the cross product vector, its magnitude, and the components individually. The magnitude doubles as the area of the parallelogram the vectors define, a handy geometric fact. The results update as you type, so you can explore how the perpendicular direction changes as you rotate the inputs. Use it for physics and engineering problems, for 3D geometry and graphics, or to check vector homework. Remember that the cross product does not commute: swapping the order of the two vectors reverses the direction of the result, flipping every sign. The calculations are exact for your inputs, with the magnitude rounded for display.

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a x b = (ay bz - az by, az bx - ax bz, ax by - ay bx). The magnitude equals the area of the parallelogram. Order matters: b x a reverses the signs.