How to prove $A\times B=\lVert A\lVert\rVert B\rVert\sin\theta N$? I want to show that if $A$ and $B$ are two linearly independent vectors, $N$ is the product of a scalar and $A\times B$ given that  $N \in \mathbb{R}^3$ is orthogonal to $A$ and $B$.

My idea to prove that is show that $N=\frac{1}{\lVert A\rVert\lVert B\rVert\sin\theta}(A\times B)$, so by hypotesis and other result
\begin{equation}
N=\alpha A+\beta B+\gamma(A\times B)
\end{equation}
for $\alpha,\beta,\gamma$ scalars, and some way I want to find linear system equation using the fact of $N$ is ortogonal to $A$ and $B$, to show that $\gamma= \frac{1}{ 
  \lVert A\rVert\lVert B\rVert \sin\theta} $ but I don't know how to do this. Do you know some hint to do this or other idea to prove this?
 A: Ok first let us define the outer product of 2 vectors and then see if we can get the usual identity for $u \times v$ . Let $u , v \in \mathbb{R}^3$ . Now for all $x \in \mathbb{R}^3$ define the function $\phi(x)=det(u,v,x)$ Now we define $u \times v$ to be the unique vector such that $<u\times v, x>=det(u,v,x)$ .Notice my $\phi$ function is a linear function. Now i can find the coordinates of  $u\times v$ considering  the standard basis and computing $< u \times v, e_i>$ using the function $\phi$.You will get $$ u\times v=
\begin{pmatrix}
u_2v_3-u_3v_2 \\
u_3v_1-v_3u_1 \\
u_1v_2-v_1u_2
\end{pmatrix}
$$.Now $||u \times v||^2=<u \times v, u\times v>=det(u,v, u\times v)=||u||^2+||v||^2-<u,v>^2$ now using the formula for the inner product and the fact that ($1-cos^2θ)=sin^2θ$ and taking the square root of $||u \times v||^2$ you will get the formula you want. To prove that this vector is normal then to $u$ and $v$ is trivial. and in general you can prove the rest of the properties from this construction.
You can use also this related answer for the computations i skipped Prove that for $x,y\in \mathbb{R}^3,\lVert x \times y \rVert^2 = \lVert x \rVert^2 \lVert y \rVert^2 - \langle x,y \rangle^2$ .
