Genesis of the following thoughts.
One usually define the following internal composition law on $\mathbb{R}^2$: $$+:\left\{\begin{array}{ccc} \mathbb{R}^2\times\mathbb{R}^2&\rightarrow&\mathbb{R}^2\\ ((a,b),(c,d))&\mapsto&(a+b,c+d) \end{array}\right..$$ One has the handy:
Geometrical interpretation 1. Let $(\mathbf{u},\mathbf{v})\in\mathbb{R}^2\times\mathbb{R}^2$, $\mathbf{u}+\mathbf{v}$ is the direct diagonal of the oriented-parallelogram formed by the vectors $\mathbf{u}$ and $\mathbf{v}$.
Moreover, the interest of $+$ is strengthened by the following algebraic:
Fact 1. $(\mathbb{R}^2,+)$ is an abelian group.
Let me try to explain what is frustrating me about $+$. Let $i\in\{1,2\}$, the $i^{\textrm{th}}$ coordinates of $\mathbf{u}+\mathbf{v}$ is only dependent upon the $i^{\textrm{th}}$ coordinates of $\mathbf{u}$ and $\mathbf{v}$ which appears to be the first natural idea one can have to define a planar vector law. However, there is no apparent good reason for $+$ to be the sole natural planar vector law and this is how I motivated myself to construct $\rtimes:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that the $i^{\textrm{th}}$ coordinates of $\mathbf{u}\rtimes\mathbf{v}$ is dependent upon both coordinates of $\mathbf{u}$ and $\mathbf{v}$ with the restriction that $\rtimes$ has sweet geometrical interpretations. I don't mind losing some algebraic properties even if it will be better if an algebraic-friendly law arises.
An attempt to define $\rtimes$ and some partial results.
Let define the following internal composition law on $\mathbb{R}^2$: $$\rtimes:\left\{\begin{array}{ccc} \mathbb{R}^2\times\mathbb{R}^2&\rightarrow&\mathbb{R}^2\\ ((a,b),(c,d))&\mapsto&(a+bc,b+ad) \end{array}\right..$$ My thought here is to weight $+$ with the relevant coordinates of $\mathbf{u}$ or $\mathbf{v}$. From there, it is relevant to introduce this other internal composition law on $\mathbb{R}^2$: $$\otimes:\left\{\begin{array}{ccc} \mathbb{R}^2\times\mathbb{R}^2&\rightarrow&\mathbb{R}^ 2\\ ((a,b),(c,d))&\mapsto&(bc,ad) \end{array}\right..$$
Observation 1. Let $(\mathbf{u},\mathbf{v})\in\mathbb{R}^2\times\mathbb{R}^2$, one has $\mathbf{u}\rtimes\mathbf{v}=\mathbf{u}+\mathbf{u}\otimes\mathbf{v}$.
Let us focus our attention on $\otimes$ and make the following:
Observation 2. Let $(\mathbf{u},\mathbf{v})\in\mathbb{R}^2\times\mathbb{R}^2$, $\mathbf{v}\otimes\mathbf{u}$ is the axially symmetric about the first bisector of $\mathbf{u}\otimes\mathbf{v}$.
One derives the following:
Fact 2. $\rtimes$ is non-commutative.
Before making some geometric observations, let us make some algebraic remarks:
Proposition 2.
$(0,0)$ is the right identity element for $\rtimes$.
Let $\mathbf{u}:=(a,b)\in\mathbb{R}^2$ such that $a\neq 0$ and $b\neq 0$, there exists a unique ${\mathbf{u}_{r}}^{-1}\in\mathbb{R}^2$ such that: $$\mathbf{u}\rtimes{\mathbf{u}_{r}}^{-1}=(0,0).$$ One has $\displaystyle{\mathbf{u}_{r}}^{-1}=-\left(\frac{a}{b},\frac{b}{a}\right)$.
However, $\rtimes$ has a nasty left behaviour.
Let us now make the geometric remarks:
Observation 3. Let $(\mathbf{u},\mathbf{v})\in\mathbb{R}^2\times\mathbb{R}^2$.
The difference of the coordinates of $\mathbf{u}\otimes\mathbf{v}$ is the area of the oriented-parallelogram formed by the vectors $\mathbf{u}$ and $\mathbf{v}$.
If $\mathbf{u}\otimes\mathbf{v}\propto (1,-1)$, then $\mathbf{u}\otimes\mathbf{v}\perp\mathbf{u}$ and $\mathbf{u}\otimes\mathbf{v}\perp\mathbf{v}$. Moreover, $\mathbf{u}$ and $\mathbf{v}$ are anticolinear.
Some interrogations.
$1.$ Is there some reference in the literature on defining such a planar vector law?
$2.$ Is there a way to temper the nasty left behaviour of $\rtimes$?
$3.$ Is there a geometrical interpretation of ${\mathbf{u}_r}^{-1}$?
$4.$ Is the converse of the second point in observation $3.$ true?
$5.$ Is there a stronger geometrical interpretation of $\otimes$ and can we derive from it a geometrical interpretation of $\rtimes$?
Any contribution or constructive remark will be greatly appreciated!
Update 1. I came up with the following results digging hjhjhj57's idea.
First, let define the following embedding of $\mathbb{R}^2$ into $\mathbb{R}^3$: $$i:\left\{\begin{array}{ccc} \mathbb{R}^2&\rightarrow&\mathbb{R}^3\\ (a,b)&\mapsto&(a,b,0) \end{array}\right.\textrm{ and }j:\left\{\begin{array}{ccc} \mathbb{R}^2&\rightarrow&\mathbb{R}^3\\ (a,b)&\mapsto&(0,a,b) \end{array}\right..$$
Fact 3. $i\left(\mathbb{R}^2\right)$ is the plane $z=0$ of $\mathbb{R}^3$ and $j\left(\mathbb{R}^2\right)$ is the plane $x=0$ of $\mathbb{R}^3$.
Now, let define the following projection of $\mathbb{R}^3$ onto $\mathbb{R}^2$: $$p:\left\{\begin{array}{ccc} \mathbb{R}^3&\rightarrow&\mathbb{R}^2\\ (a,b,c)&\rightarrow&(a,b) \end{array}\right.$$ One can interpret $p$ has the projection onto the plane $z=0$. Let define the following transformation of $\mathbb{R}^2$: $$\sigma:\left\{\begin{array}{ccc} \mathbb{R}^2&\rightarrow&\mathbb{R}^2\\ (a,b)&\mapsto&(b,a) \end{array}\right..$$ $\sigma$ is the axial symmetry about the first bissector i.e. $y=x$. From there, one can make the following:
Observation 4. Let $(\mathbf{u},\mathbf{v})\in\mathbb{R}^2\times\mathbb{R}^2$, one has: $$\mathbf{u}\otimes\mathbf{v}=p\left(i(\sigma(\mathbf{u})\right)\times j(\mathbf{v})).$$ In other words, $\mathbf{u}\otimes\mathbf{v}$ is the projection onto $z=0$ of the cross-product of the axially symmetric about the first bissector of $\mathbf{u}$ seen into the plane $z=0$ with $\mathbf{v}$ seen into the plane $x=0$.
Proof. For all $\mathbf{u}:=(a,b),\mathbf{v}:=(c,d)\in\mathbb{R}^2$, one has $\begin{pmatrix}b\\a\\0\end{pmatrix}\times\begin{pmatrix}0\\c\\d\end{pmatrix}=\begin{pmatrix}bc\\ad\\-bd\end{pmatrix}.$ $\Box$
One can also make the following observation: $$\begin{pmatrix}b\\a\\d\end{pmatrix}\times\begin{pmatrix}b\\c\\d\end{pmatrix}=\begin{pmatrix}bc\\ad\\0\end{pmatrix}$$ , which probably leads to another geometrical interpretation of $\mathbf{u}\otimes\mathbf{v}$. The strong point of this last identidy is that while projecting the result onto the plane $z=0$ to recover $\mathbf{u}\otimes\mathbf{v}$, one only loses a coordinate equal to $0$ and thus does not lose information. However, the embeddings of $\mathbb{R}^2$ into $\mathbb{R}^3$ used are non-canonical and they are depends of the coordinates of $\mathbf{u}$ and $\mathbf{v}$.
How can these observations be used to answer my interrogations?