Any help will be appreciated!

I am solving Laplace's Equation in 3-Dimensions using superposition.

The problem is that I don't know how to use orthogonality to solve for the constant when you have two Eigen functions.

This is my attempt to get a solution for v(x,y,z).

The solution using the given homogeneous boundary conditions comes out to be:

where and

The remaining boundary condition is:

Which gives:

How do I use orthogonality to find K when I have two eigen functions in the solution?