Cross Product Definition and Properties

Calculus 3 · Axiom Academy

One product of two 3D vectors that returns a third vector — perpendicular to both, as long as the parallelogram they span. Given and , the cross product is the symbolic determinant with across the top row. Expanding it along that row — a cofactor expansion — is exactly what the animation walks through: each basis vector's coefficient is the minor left after you delete its row and column. Symbolic determinant with the basis vectors on top Expanded, this is the component formula for 2. Direction: Perpendicular, by the Right-Hand Rule The result is perpendicular to the whole plane of and : dot it with either input and you get zero. Which of the two perpendicular directions? Point your right hand's fingers along , curl toward , and your thumb gives . In the animation the result vector rises out of the plane — then, when the order swaps to , it flips to point the other way. — the result never leans toward the first vector. — nor toward the second. It stands normal to the plane. — same length, opposite way. This is anti-commutativity. A single perpendicular direction like this only exists in three dimensions — the plane has exactly one normal line. Take (along ) and (along ), both in the xy -plane. Then — pointing straight up the z -axis, out of their plane, exactly as the right-hand rule predicts. Swap them and you get . 3. Magnitude: the Area of the Parallelogram

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