I only saw constructions where we posit the different relations such as $\hat\imath\times\hat\jmath=\hat k$ etc, but I want to do is this.
I want a vector yielding operation $\vec w=\vec u\times\vec v$ that would give me:
- $\vec w\cdot\vec u=0$
- $\vec w\cdot\vec v=0$
- $|\vec w|=|\vec u||\vec v|\sin\left(\angle(\vec u,\vec v)\right)$
So if $\vec w=(a,b,c)$, $\vec u=(d,e,f)$, and $\vec v=(g, h, i)$, I have:
- $ad+be+cf=0$
- $ah+bh+ci=0$
- $a^2+b^2+c^2=(d^2+e^2+f^2)(g^2+h^2+i^2)-(dg+eh+fi)^2$
I squared $(3)$ and got the sine from $\vec u\cdot\vec v$.
What next, though?
Frankly, I don't like to think that we have the norm of the vector product giving us the area of the parallelogram described by both vectors be just a happy coincidence.
Edit:
I can expand $3$ to get $|\vec w|^2=(ei-hf)^2+(di-gf)^2+(dh-eg)^2$, which is literally the norm of the usual vector product, but I haven't at all used $1$ and $2$. It seems too arbitrary to just assume $(a,b,c)=(ei-hf,di-gf,dh-eg)$, especially knowing that I have ordered them in that specific manner because of my prior knowledge of the $\vec w$'s coordinates.