Find the equation of the plane passing through the points $A (0, 1, 2)$ , $B (2, - 1, 1)$ and $C (1, 3, 0)$ .

hard 4 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 $2 x + y + 2 z - 5 = 0$ , i.e., $2 x + y + 2 z = 5$ .

Step-by-step solution

Compute vector $AB = B - A = (2, - 2, - 1)$ .
Compute vector $AC = C - A = (1, 2, - 2)$ .
Find normal vector by cross product $n = AB \times AC = (6, 3, 6)$ .
Simplify normal by dividing by 3: $n = (2, 1, 2)$ .
Use point $A (0, 1, 2)$ in plane equation $n \cdot ((x, y, z) - (0, 1, 2)) = 0$ .
Dot product gives $2 (x - 0) + 1 (y - 1) + 2 (z - 2) = 0$ .
Expand: $2 x + y - 1 + 2 z - 4 = 0$ .
Simplify to $2 x + y + 2 z - 5 = 0$ , i.e., $2 x + y + 2 z = 5$ .

Common mistake: Failing to compute cross product correctly

Concept tested: plane through three points