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At the moment, I'm following this page to create a physics engine.

As per that page, I need to cross angular velocity by a distance vector to get relative torque and such, but I'm not sure how (in a 2D plane), you can cross a scalar(angular velocity), and a vector(such as the distance vector). I've assumed so far that you multiply the magnitude, but obviously this isn't correct, as I continue the debugging of my flimsy code. What am I missing?

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The cross product is not defined between a vector and a scalar, so it means nothing (it would be a malformed operation, or perhaps a mis-interpretion of a simple vector scaling operation).

In the link you provided, angular velocity is a vector when it is used in the equation that involves its presence as an operand of the cross product:

Here we are regarding the angular velocity as a 3 dimensional vector perpendicular to the plane, so that the cross product is calculated as

(Emphasis mine.)

This is in the section "Physics of Collision for Rigid Bodies in 2 Dimensions."

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