Cross Product definition

Question

Why is the cross product defined as the area of the parallelogram created by making the vectors $\vec{a}$ and $\vec{b}$ it’s sides, multiplied by the unit vector $\vec{n}$ which is orthogonal to $\vec{a}$ and $\vec{b}$?

Answer 1

There are various equivalent definitions, and they arise because the same combination comes up again and again in practice and it is convenient to have a shorthand notation for it.

If you read the Feynman Lectures in Physics, for example, Feynman explains how the vector formulation of physical law encodes the symmetry of laws under translation and under rotation. Since physics has such symmetries, only the combinations of physical properties which have such symmetry can be part of physical laws. It is therefore almost inevitable that the simplest combinations which have this symmetry will appear in some form. The cross product has such symmetry.

Note also and importantly: Your definition is incomplete, because there are two unit normals to a plane, pointing in opposite directions, and you haven't specified how to choose.

Answer 2

Good question! Basically, the definition of the cross product is arbitrary and somewhat silly.

For many purposes in math and physics, it's useful to have some measure of how "similar" two vectors are. There are a few ways to define it; the most common one is what you know as the dot product. But this isn't invertible, and it's missing some information: namely, how "different" the vectors are, which can be represented by the parallelogram between them.

This parallelogram can be thought of as a "directed area". Just like how a vector is a "directed length", this is an area with a direction to it (where the direction might be "from +X toward +Y", or "from +Y toward -Z", or whatever). Hermann Grassmann, who first came up with this, called it a "bivector". And it turns out these bivectors are very good for representing certain things in physics, such as angular momentum and magnetic fields.

But to make a long story short, the physicists didn't like directed areas. They wanted everything to be vectors. And since a bivector in three dimensions has three orthogonal components (XY, XZ, YZ), and a vector in three dimensions has three orthogonal components (X, Y, Z), they decided that all bivectors should now be written as vectors: Z = XY, X = YZ, Y = ZX. And thus the modern cross product was born.

And this works! …mostly. It enforces a "handedness" on the universe, which is why you have to use the right hand rule and not the left hand rule to take cross products. It means that some vectors flip their direction when you look in a mirror, while others don't. And it only works in three dimensions: in two dimensions, for example, a bivector has only one orthogonal component (XY), while a vector has two. And in four dimensions, a bivector has six components, while a vector has four. But for the simple three-dimensional case, it's "good enough", as long as you don't look too closely.

(If you're interested in bivectors and multivectors and alternatives to the cross product, look up the "wedge product", and "Clifford algebra" aka "geometric algebra". The alternatives might make more sense to you.)