I'll try to give you a very basic geometric explanation about the 2 vector products.
1. The result of the dot product of two vectors is a real number.
The result of the cross product of two vectors is a vector.
2. Dot product:
If you have 2 vectors then the dot product is:
Since is the length of and is the length of the product
is the length of projected perpendicular on the .
And since the product of 2 numbers can be represented as the area of an rectangle the shaded rectangle represents the value of the dot product.
3. Cross product.
two vectors which are not-collinear ( don't have the same direction) span a plane P. The cross product of and gives a new vector which is perpendicular to both vectors that means the new vector is the normal vector of the plane (a plane has only one normal vector!):
The length of is calculated by:
which describes the area of a parallelogram with and as it's sides.