Physics!!!?

Question

1) Why in math do we replace the cross product between radius and Force to
calculate Torque, with the sine function? Why can't we use cosine instead?

Answer 1

If you replaced sine with cosine, you'd have the formula for the dot product instead of the cross product. The cross product is greatest when the two vectors are perpendicular to each other, that is, when sine is maximized. That's why the cross product is used for things involving lever arms, when a vectors effective distance from the origin of another vector is relevent. On the other hand, the dot product is greatest when cosine is maximized, meaning that the two vectors are colinear. That's why the dot product is used when one vector acts along the line of another vector, such as mechanical work.