Find the equation of the plane that passes through the points $(0, 1, 2)$ , $(2, - 1, 1)$ , and $(1, 3, 0)$ .

easy 2 marks Class 12 · Mathematics · Three Dimensional Geometry

Answer

2 x + y + 2 z = 5

The correct answer is $2 x + y + 2 z = 5$ — Simplify to obtain $2 x + y + 2 z = 5$ .

Step-by-step solution

Form two vectors in the plane using the given points: $v_{1} = (2, - 1, 1) - (0, 1, 2) = (2, - 2, - 1)$ and $v_{2} = (1, 3, 0) - (0, 1, 2) = (1, 2, - 2)$ .
Compute the cross product $n = v_{1} \times v_{2} = (6, 3, 6)$ .
Simplify the normal vector by dividing by $3$ , giving $n = (2, 1, 2)$ .
Use the point–normal form $2 (x - 0) + 1 (y - 1) + 2 (z - 2) = 0$ to write the plane equation.
Simplify to obtain $2 x + y + 2 z = 5$ .

Common mistake: Forgetting to simplify the normal vector before writing the equation.

Concept tested: Plane through three points

Published 7 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked