How do the dot product and the cross product resemble (or differ from) the ordinary product of two numbers?

The dot product is used to determine the resultant vector from the given component vectors. The ordinary product just simply multiplies the magnitude of the vector. Basically, the dot product incorporates matrices. You can determine the resultant vector by finding the determinant. Hence, the dot product involves a lot of steps.

What do you mean by that? like the contents? the thc level?