Linear Algebra: Cross Product w/ Matrices

Question

Let v = $\begin{bmatrix} v_1\\ v_2 \\ v_3 \\ \end{bmatrix}$. Which matrix A implements the transformation:
$x = \begin{bmatrix} x_1\\ x_2 \\ x_3 \\ \end{bmatrix}$ $\rightarrow $ $v\times x = \begin{bmatrix} v_2x_3 - v_3x_2\\ v_3x_1 - v_1x_3 \\ v_1x_2 - v_2x_1 \\ \end{bmatrix}$

So I've been thinking about this geometrically, and can't really wrap my head around it: is it just the 3x3 identity matrix? This would prove to be a perpendicular matrix to any two given vectors. Nevertheless, the question itself confuses me: how does a matrix itself implement a cross product? Does it mean to just give an example of a matrix that serves as an intermediary to give the answer?

Answer 1

It's the matrix$$\begin{bmatrix}0&-v_3&v_2\\v_3&0&-v_1\\-v_2&v_1&0\end{bmatrix},$$as you can check.

Answer 2

Note that

$$\mathbf{a} \times \mathbf{b} = \begin{bmatrix}\,0&\!-a_3&\,\,a_2\\ \,\,a_3&0&\!-a_1\\-a_2&\,\,a_1&\,0\end{bmatrix}\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix}$$

Answer 3

The matrix $A$ implements the cross product of a fixed vector $v$ with a variable vector $x$. You’ve already got a formula for this cross product in the question itself. Simple extract the coefficients on the right-hand side into a matrix.