Cross Product Calculator

Point the fingers of your right hand in the direction of A, then curl them toward B through the smaller angle. Your thumb now points in the direction of A x B. That direction is always perpendicular to both A and B. If you curl from B toward A instead, the thumb flips, which is why B x A = -(A x B). The calculator reports the unit vector of A x B so you can read off the direction without doing the hand trick.

The cross product is anti-commutative: swapping the inputs flips the sign of every component, so B x A = -(A x B). The magnitudes match, but the result points the opposite way. Geometrically it is the same right-hand-rule story: curling from B to A puts your thumb on the other side of the A-B plane. Dot products and ordinary multiplication are commutative; the cross product is one of the few common operations where order matters.

The ordinary cross product used in geometry and engineering maps two 3D vectors to one perpendicular 3D vector. In 2D, the determinant a1*b2 - a2*b1 gives a signed scalar area. Higher-dimensional analogues use exterior products, and an exceptional binary vector cross product also exists in 7D. This calculator implements only the standard 3D operation.

|A x B| equals the area of the parallelogram whose sides are A and B. Half that value is the area of the triangle with the same two sides, which is why surveyors, CAD tools, and physics engines use the cross product for area work. The calculator reports both: areaOfParallelogram = |A x B| and areaOfTriangle = |A x B| / 2. When |A x B| = 0, the two vectors are collinear and the parallelogram has collapsed to a line segment.

Torque is r x F (the lever arm crossed with the force), so a wrench applies the same torque whether you turn it clockwise or counterclockwise depending on which way r points. Angular momentum is r x p (position crossed with linear momentum). The magnetic force on a moving charge is q(v x B). In 3D graphics, a triangle's surface normal is (V2 - V1) x (V3 - V1), normalized. Anywhere a quantity needs a perpendicular direction and a magnitude that scales with the area or the sine of an angle, the cross product is the tool.