Vector Dot and Cross Products

OK I'm pretty new to Vectors and I'm missing something very basic here.

Any help would be appreciated.

I can recite equations on both Dot and Cross products until I am blue in the face.

I can even understand where the Dot product comes from in terms of proving it using triangles and the Cosine Rule.

The question I'm struggling to answer is exactly what are these products showing me and why in layman's terms?

I've spent 2 days solidly reading everything I can on this and haven't found a satisfactory answer.

My best guesses are as follows:-

1) For the Dot product, you are essentially applying two forces and working out a combined effect of both of them. Clearly the maximum effect is if both of these forces are in the same direction. As you move one of them away you can use triangles to work out the effective "strength" of the second force which will be between 0 and 1 times it's force magnitude.

This answer can't be right because at 90 degrees, the Dot Product is zero but if you had two forces at 90 degrees of the same magnitude clearly there would be movement at 45 degrees. What have I missed here?

2) I suspect, the Cross Product is showing the resultant rotation from two forces acting on each other. Is this correct? If so, this may well help explain why we use sine theta because that would be the force line of force one exerted on force two with a rotation about the point where the vectors tails are. How does this work with displaced vectors though is they are NOT tail to tail?

I'm sure I'm miles out with my thoughts but I don't want anyone thinking I haven't at least had a go at this myself.

I particularly want to know this because my interest is in Electromagnetics and I need to userstand this before I start to move onto Div, Curl an d other frightening things :)