How to mathematically determine if the magnitude of a cross product is up/down(positive/negative?)?

Question

So, I'm a newbie at complex vector math. I'm working on a 2D physics engine, and my issue is, with angular acceleration from torque, is it supposed to be positive or negative? I understand the right hand rule vaguely, but not how it translates into math. A google search didn't yield anything, and I'm quite clueless.

Answer 1

If you are computing everything component-wise (meaning you need to have defined an orthonormal basis {$\hat{x},\hat{y},\hat{z}$}), as shown below (taken from here):

remains to just perform the cross product of the force $\mathbf{F}$ and $\mathbf{r}$ of your object, meaning: $$\mathbf{F}\times \mathbf{r} =(F_y r_z-F_z r_y)\hat{x}+(F_z r_x-F_x r_z)\hat{y}+(F_x r_y -F_y r_x)\hat{z} $$

Which is the torque vector with the above components.