Pick any two points on the line.
• if x = 1, then y = 4 and z = 3
• if x = -1, then y = 7 and z = 2
Observe that P is not on the given line.
The vector from (1, 4, 3) to P points in the same direction as
(0, 5, 2) - (1, 4, 3) = (-1, 1, -1)
and the vector from (-1, 7, 2) to P points in the same direction as
(0, 5, 2) - (-1, 7, 2) = (1, -2, 0)
The cross product of these vectors is
(-1, 1, -1) × (1, -2, 0) = (-2, -1, 1)
and is normal to the plane we want to find. Since the plane contains P, the equation of the plane is
(-2, -1, 1) • (x - 0, y - 5, z - 2) = 0 ⇒ 2x + y - z = 3