The Cross Product

The second way to multiply vectors produces a new vector, one that stands perpendicular to both of the originals. Torque, angular momentum and magnetic force are all built this way.

Magnitude#

The cross product (or vector product) $\vec a\times\vec b$ is a vector whose size is

Draw the two vectors tail to tail and you see two angles between them, one less than 180° and one more (they add to 360°). The θ here is always the smaller one, so $\sin\theta \ge 0$ and the magnitude is never negative.

Geometrically, $ab\sin\theta$ is the area of the parallelogram spanned by the two vectors (base $a$ times height $b\sin\theta$). Parallel or anti-parallel vectors span no area, so their cross product is zero.

Direction: the right-hand rule#

$\vec a\times\vec b$ points perpendicular to the plane containing $\vec a$ and $\vec b$. That leaves two choices, up or down, and a convention picks one:

Push $\varphi$ past 180° in the lab above. The smaller angle is now measured the other way round, your fingers curl the opposite way, and the product flips downward. Hit “swap order” and the same thing happens. Reversing the order of a cross product reverses its direction:

It does distribute over addition, but it is not associative:

Right-handed axes#

Once you choose x and y axes, the z-axis must be perpendicular to both, but it could point either way. We always choose it so that $\hat\imath\times\hat\jmath = \hat k$. Such a frame is called right-handed, and everything else follows from the cycle:

Cross product from components#

Expand $(a_x\hat\imath + a_y\hat\jmath + a_z\hat k)\times(b_x\hat\imath + b_y\hat\jmath + b_z\hat k)$ into nine terms and use the cycle above. Three terms vanish (a unit vector crossed with itself) and the rest pair up: