Like the dot product, the cross product has some nice properties.
It's easy to see that 
.
Also if two vectors are going in the same direction, the cross product
is zero. If the vectors are perpendicular then 
so that the magnitudes just multiply. Let's work out some of the
cross products between unit vectors:
So now we should be able to work out the cross product in cartesian coordinates by multiplying out all of the components. It's a lot of work and the answer is
The component formula is usually the simplest way to work out problems involving cross products.