Direction solution to move a point along a 3D plane?

Question

Game development

Good afternoon.
There is a 3D plane equation and some point on it
needs to be shifted along the plane by X and Y, i.e. so that it remains to belong to the plane (as if the local coordinate system is obtained ... or not).
I don't know how to step up.
Please advise the direction of the solution. I tried to start from the normal and the multiplication of vectors - it did not work, because apparently there is no understanding ...

Answer 1

Most likely, it will be convenient to find the coordinates of the basis unit vectors (OX, OY) on this plane. Then the displacement (dx, dy) will be expressed in terms of them a_new = a + dx * OX + dy * OY.

Answer 2

If I understood you correctly, then you have an equation of the plane (ax+by+cz+d=0), a point (x1,y1,z1), which must be reduced to the form (x2,y2,?), in which it will be belong to the plane?
In this case, z2 = ( -ax2-by2-d ) / c

Answer 3

And what exactly is the equation for the plane?
What kind of movement needs to be implemented, where will the point move?
Is the plane initially given by an equation or a set of points? If there are three points that do not lie on one straight line, they will define two vectors that are uniquely coordinates for this plane.
It will not be difficult to compose an equation in order to obtain the internal coordinates of the point in the plane from the three-dimensional coordinates of the point. The next question is what kind of movement should be implemented.