Cross Product

The definition of the cross product is:


... where ε^{ijk} is the Levi-Civita Symbol.

Where does the angle come from?

First, there is an identity connecting the cross product to the scalar product:


So the identity is:


If the vectors c and a are the same and the vectors d and b are the same, it follows:

|a⃗⨯b⃗|^2=|a⃗|^2⋅|b⃗|^2⋅(1-(\cos φ)²)
|a⃗⨯b⃗|^2=|a⃗|^2⋅|b⃗|^2⋅(\sin φ)²
|a⃗⨯b⃗|=|a⃗|⋅|b⃗|⋅\sin φ

Hence the angle in the cross product comes from the angle in the scalar product:

Scalar Product

Given a triangle with sides a and b and c=b-a, the Law of Cosines says:

|c⃗|²=|a⃗|²+|b⃗|²-2⋅|a⃗|⋅|b⃗|⋅\cos φ
|b⃗-a⃗|²=|a⃗|²+|b⃗|²-2⋅|a⃗|⋅|b⃗|⋅\cos φ

On the other hand, using only the scalar product distributivity:


So since (b⃗-a⃗)⋅(b⃗-a⃗)=|b⃗-a⃗|², it follows that:

2⋅a⃗∙b⃗=2⋅|a⃗|⋅|b⃗|⋅\cos φ
a⃗∙b⃗=|a⃗|⋅|b⃗|⋅\cos φ