geometry – Given a plane and a point on it. How does the point change when its perpendicular vector is changed?

Suppose I have a plane with a perpendicular vector (0,0,1) and a point on it (1,0,0) or any point (x, y, z). Suppose I tilt the plane so that the perpendicular vector becomes (a, b, c). Describe the transformation of (x, y, z) into new coordinates (X, Y, Z).




I need a rule for that

I can use the original perpendicular vector (x, y, z)
new perpendicular vector (X, Y, Z)
old point (x, y, z) to understand

new coordinates of the point (X, Y, Z)

a simple formula without proof would also for the moment

Please help