Plane

After rotation of coordinate axes about the origin in the positive direction through on angle , the new coordinates are have relation with older coordinates is
, where


so, becomes



On comparing, we get
and

Step 1: Section formula in 3D Using the section formula, the point dividing and in the ratio is: Solving for the given plane equation, we find:

Step 1: Finding the equation of the required plane
The given plane equation is: Since the required plane is parallel to this plane, its equation must be of the form: Since the plane passes through , substituting these values: Thus, the equation of the required plane is: Step 2: Finding the foot of the perpendicular
The formula for the foot of the perpendicular from to the plane is: Substituting , , , and the given point : Since the foot of the perpendicular lies on the plane , substituting these values: Expanding: Step 3: Finding the coordinates
Thus, the foot of the perpendicular is:

Step 1: Finding the perpendicular distance The perpendicular distance from a point to the plane is given by: Substituting the given values: Step 2: Computing Given , solving using the foot of the perpendicular formula, we find: