The formula
a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)
What it means
Unlike the dot product, which returns a number, the cross product returns a vector. It points perpendicular to both inputs, in the direction given by the right-hand rule, and its length equals the area of the parallelogram the two vectors define.
How to do it by hand
- Both vectors need exactly three components
- Each output component is a 2×2 determinant of the other two
- Check: the result dotted with either input should give zero
- Its magnitude is the parallelogram area; half of it is the triangle
What is worth knowing
The cross product exists only in three dimensions, and that is not an oversight — it is a genuine peculiarity. In three dimensions the space of directions perpendicular to two given vectors is exactly one-dimensional, so 'the' perpendicular is well defined up to sign. In four dimensions it is a plane, and there is no single answer. It is also anticommutative: a × b = −(b × a), so swapping the order flips the vector. In physics it defines torque, angular momentum and the magnetic force on a moving charge, all of which point 'sideways' for exactly this reason.