How do I find the angle between a vector and a plane in cartesian form?

First of all, you should find the angle between the vector and the normal of the plane using the rule cosX=a.b/modamodb. The normal can be found easily from the cartesian equation of the plane. For example, the cartesian plane 2x+5y-3z=6 has normal (2  5  -3). 

The angle that you find using this method is the angle between the vector and the normal. Since we wish to find the angle between the vector and the plane, we must therefore subtract our result from 90 degrees to find the correct one.