Cross Product Calculator

The cross product is an operation on two vectors in three-dimensional space. Its result is another vector rather than a single scalar. That result points perpendicular to both input vectors, so it is especially useful whenever a problem involves orientation, a surface normal, rotation, torque, or an area in space. The order matters: reversing the vectors reverses the direction of the answer. In other words, A cross B is the negative of B cross A. This calculator applies the standard component formula directly. For vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz), the first result component is AyBz minus AzBy. The second is AzBx minus AxBz, and the third is AxBy minus AyBx. These three signed differences are equivalent to expanding the familiar determinant built from the unit directions i, j, and k. Enter integers, decimals, or negative values; the calculator evaluates each component and presents the resulting ordered triple. The magnitude of the cross product has a geometric meaning. It equals the area of the parallelogram spanned by the two vectors. Half that magnitude is the area of the triangle formed by the same vectors. If the result is the zero vector, the inputs are parallel, anti-parallel, or one vector has zero length. The direction of a nonzero result follows the right-hand rule: curl the fingers of your right hand from A toward B, and your thumb points along A cross B. Cross products appear throughout physics, engineering, computer graphics, and linear algebra. Torque is the cross product of a position vector and force. Angular momentum uses position and momentum. Graphics software computes surface normals from two edges of a polygon, while geometry problems use the magnitude to find areas. Because small sign mistakes are common when calculating by hand, this tool is useful for checking each component quickly. The displayed formula also makes the coordinate order explicit, helping you compare the result with a manual determinant expansion.