I just want to know how the geometric, algebraic, and trigonometric definitions of the dot and cross products of two vectors relate to each other.

The dot product is defined as both the sum of squares of components of two vectors and also the product of their lengths times cos(θ). I also know that the cross product is the determinant of the matrix with unit vectors i,j, and k in the first row, the components of a in the second, and that of b in the third and also the product of their lengths times sin(θ) times a unit vector n that is perpendicular to both a & b.

But how does:

and the determinant of

I already understand the geometric definition via example (Work = Force*Distance and Torque), and you don't have to explain it to me algebraically; as long as the fundamental idea is explained, any way is fine, thanks.