This might be a good start, though depending on your level some of the material might be beyond you. It could serve as the basis for more searching, in any event.
Can anyone explain what the motivation is behind the vector cross product?
Why have the sine of the angle between the vectors as a multiplicand?
What makes the product another vector at right angles to the multiplicand vectors?
I mean, the guy who first decided that it should be like that must have had some reasons for it.
I can intuitively feel the idea behind dot product but this cross product baffles me.