Question

Please help me with the below question.

Answer 1

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