The cross product

Where the dot product collapses two vectors into a number, the cross product builds a new vector out of them. It exists only in three dimensions, and it is defined so that a × b is perpendicular to both a and b, with length |a||b| sin θ — the area of the parallelogram the two vectors span.

That sin θ is the mirror image of the dot product's cos θ, and it flips the behaviour completely. The cross product is largest when the vectors are perpendicular and zero when they are parallel, because parallel vectors span a degenerate parallelogram with no area at all. So a zero dot product means perpendicular, and a zero cross product means parallel — two complementary tests worth keeping straight.

The one property that trips people is that order matters: a × b = −(b × a). Swapping the operands does not change the length but reverses the direction, which the right-hand rule makes concrete — curl your fingers from the first vector to the second and your thumb gives the result. This handedness is not a quirk of notation. It is why torque, angular momentum and the magnetic force on a moving charge all have a direction that has to be stated with a rule about hands rather than derived from magnitudes alone.