Vector product in two dimensions

Question

Hello I have a question related to the product of two vectors in two dimensions. I have an equation and therein a product of two vectors appears. The result must be a vector, since it is a force equation. Now I am confused, what kind of multiplication is used there. I have the feeling it is a geometric product, but I am not sure what to do to check it. My idea was just to state that there exist no other multiplication in 2D of two vectors resulting a vector. Is this true?

Answer 1

Not sure I understand it completely, but here are some examples of multiplication. Please indicate whether you recognize the structure or anything like that. If you cannot recognize anything, please add more informatino.

First of all, we have the dot product, denoted by $\cdot$

Let $\mathbf{x} = \begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}, \quad \mathbf{y} =\begin{bmatrix}y_1\\y_2\\y_3 \end{bmatrix} $, then we have $\mathbf{x}\cdot\mathbf{y} = x_1y_1 + x_2y_2 + x_3y_3$

Also, we have the cross product (defined in $\mathbb{R}^3$), denoted by $\times$

Let $\mathbf{x} = \begin{bmatrix}x_1\\x_2\\x_3 \end{bmatrix}, \quad \mathbf{y} =\begin{bmatrix}y_1\\y_2\\y_3 \end{bmatrix} $, then we have $\mathbf{x}\times \mathbf{y} = \begin{vmatrix} i & j & k \\ x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \notag \end{vmatrix} = (x_2y_3 - x_3y_2) i + (x_1y_3 - x_3y_1)j + (x_1y_2 - x_2y_1)k$ They do not result in matrices, but maybe you recognize something

Answer 2

Although I cannot be sure without more context whether this is relevant for you, I have already seen this: when people talk about the vecor product of two $2$-dimensional vectoras $(a,b)$ and $(c,d)$, what they have in mind is $(a,b,0)\times(c,d,0)$.