Determine the Cartesian equation of the plane that passes through the points (1, 4, 5) and (3, 2, 1) and is parallel to the plane 2x – y + z + 1 = 0.

Question

Determine the Cartesian equation of the plane that passes through the points (1, 4, 5) and (3, 2, 1) and is parallel to the plane 2x – y + z + 1 = 0.

Answer 1

A plane such that does not exist.

Let us assume that the above statement is not true.

That is let us assume that there exists a plane containing the two given

points parallel to the given plane.

Then the line passing through these two points would be orthogonal to the normal vector of the given plane.

The normal vector of the given plane is n =< 2, -1 ,1 >

Also the directing vector defined by the two given points is u = <2, -2, -4 >

But the dot product of the two vectors n • u = 4 -2 -4 ≠ 0 .Contradiction.

We arrived at a contradiction because we assumed that such a plane exist.

Therefore the supposition is not true.

Therefore a plane as that does not exist