Find the equation of plane passes through the point ` A ( 1,2,1) ,B ( 2,3,1) and C( 2,1,0) ` Also find the distance of the point (1,1,1)

Answer

To find the equation of the plane passing through the points \( A(1, 2, 1) \), \( B(2, 3, 1) \), and \( C(2, 1, 0) \), and to calculate the distance from the point \( P(1, 1, 1) \) to this plane, we can follow these steps:

### Step 1: Find the vectors in the plane

1. **Calculate the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \)**:
- \( \overrightarrow{AB} = B - A = (2 - 1, 3 - 2, 1 - 1) = (1, 1, 0) \)
- \( \overrightarrow{AC} = C - A = (2 - 1, 1 - 2, 0 - 1) = (1, -1, -1) \)