as i understand you r probably trying to apply a affine transformation of the form Av+b, where A is an orthogonal isometry and b is some vector. assuimng that your orginal plane is aligned with the x-axis and you need to tilt its y-axis (called j vector) to (say) v direction. v is auusmed normalized. let us call the vector perp to the plane (j,v) as n.

find the matrix for this rotation -- axis is n and j rotates to v.

this matrix is your answer.