# Why is cross product not commutative?

Why, conceptually, is the cross product not commutative? Obviously I could simply take a look at the formula for computing cross product from vector components to prove this, but I'm interested in why it makes logical sense for the resultant vector to be either going in the negative or positive direction depending on the order of the cross operation. I don't have any formal experience in linear algebra, so I would appreciate if an answer would take that into account (I'm merely learning vectors as part of a 3D game math education).

• There are two ways of looking at such a thing: cross product is actually just a binary operation and the formula is, in fact, an axiom. However, axioms are defined so they may be useful and various concepts in physics, such as in electricity and magnetism or in kinetics seem to correspond with this notion.
– Jas
Commented May 28, 2015 at 18:30

The magnitude of the resulting vector is a function of the angle between the vectors you are multiplying. The key issue is that the angle between two vectors is always measured in the same direction (by convention, counterclockwise).

Try holding your left thumb and index finger in an L shape. Measuring counterclockwise, the angle between your thumb and index finger is roughly 90 degrees. If you measure from your index finger to your thumb (still must be done counterclockwise!) you have roughly a 270 degree angle.

One way to calculate a cross product is to take the determinant of a matrix whose top row contains the component unit vectors, and the next two rows are the scalar components of each vector. Changing the order of multiplication is akin to interchanging the two bottom rows in this matrix. It is a theorem of linear algebra that interchanging rows results in multiplying the determinant by -1.

Since two vectors are perpendicular to any two non parallel vectors, and these vectors are in opposite directions, it makes sense to decide which one is to be the result of the cross product. So the convention was adopted to follow a right hand triad, as opposed to a left hand triad.

Because $\vec{a}$, $\vec{b}$ and $\vec{a}\wedge\vec{b}$ are a right hand triple of vectors.

And, if $(\vec{a},\vec{b},\vec{c})$ is a right hand (ordered) triple of vectors, then also $(\vec{b},\vec{a},-\vec{c})$ is, while $(\vec{b},\vec{a},\vec{c})$ is not. Indeed not only cross product is not commutative, further it is anticommutative.

See, also, the cross product tag wiki.

• Yes. Part of the historical development was inspired by electromagnetism where the difference between right- and left-handed systems physically matter. (E.g., the Lorenz force $\vec F$ on an electric charge $q$ with velocity $\vec v$ in a magnetic field $\vec B$ is given by $\vec F = q\vec v \times \vec B$.) Commented May 28, 2015 at 18:39