so i'm trying to get
(-1,2,-1) and (1,1,-2) multiplication into a new vector
book says (5,-3,1)
unfortunately it showed us how to do 2x2 and 3x3 matrixes and I learned how to apply those concepts further, but it never showed anything like this (there is a formula for it in the book but it i want to know if I can use a matrix to multiply this - the formula seems a bit cumbersome to memorize)
Take the determinant of a matrix with three rows, being the row $(i,j,k)$, vector 1, and vector 2.
Taking the determinant across the top row will yield some $\alpha i + \beta j + \gamma k$ which obviously is the vector $(\alpha,\beta,\gamma)$
I am sure you can recognize that cumbersome formula is simply an application of this.
Cross Product: $a = (a,b,c), b = (d,e,f)$
$$ a \times b =\left| \begin{align}& i&j&&k\\&a & b&&c\\&d&e&&f \end{align} \right|$$ $i(bf-ec) - j(af-dc) + k(ae-db)$
This generates an orthogonal vector to both vectors one and two.
Try to use the $\epsilon_{\jmath k \ell}$ symbol.
$$\big[ \textbf{v} \times \textbf{w} \big]_\jmath = \sum_{k,\ell} \epsilon_{\jmath k \ell} v_k w_\ell$$
