Consider the plane passing through the points `A(2,2,1),B(3,4,2) and C(7,0,6)`
What are the direction ratios of the normal to the plane

Options

A. `lt1,0,0gt`
B. `lt0,1,0gt`
C. `lt1,0,-1gt`
D. None of these

Answer

To find the direction ratios of the normal to the plane passing through the points \( A(2,2,1) \), \( B(3,4,2) \), and \( C(7,0,6) \), we will follow these steps:

### Step 1: Find the vectors AB and AC
First, we need to find the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \).

- The vector \( \overrightarrow{AB} = B - A = (3-2, 4-2, 2-1) = (1, 2, 1) \)
- The vector \( \overrightarrow{AC} = C - A = (7-2, 0-2, 6-1) = (5, -2, 5) \)

### Step 2: Calculate the cross product of vectors AB and AC
The direction ratios of the normal to the plane can be found by calculating the cross product of the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \).

Let:
\[ \overrightarrow{AB} = (1, 2, 1) \]
\[ \overrightarrow{AC} = (5, -2, 5) \]

The cross product \( \overrightarrow{AB} \times \overrightarrow{AC} \) is given by the determinant:
\[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 1 \\ 5 & -2 & 5 \end{vmatrix} \]

### Step 3: Calculate the determinant
Calculating the determinant, we have:
\[ \hat{i} \begin{vmatrix} 2 & 1 \\ -2 & 5 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 1 \\ 5 & 5 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 2 \\ 5 & -2 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. \( \begin{vmatrix} 2 & 1 \\ -2 & 5 \end{vmatrix} = (2 \cdot 5) - (1 \cdot -2) = 10 + 2 = 12 \)
2. \( \begin{vmatrix} 1 & 1 \\ 5 & 5 \end{vmatrix} = (1 \cdot 5) - (1 \cdot 5) = 5 - 5 = 0 \)
3. \( \begin{vmatrix} 1 & 2 \\ 5 & -2 \end{vmatrix} = (1 \cdot -2) - (2 \cdot 5) = -2 - 10 = -12 \)

Putting it all together, we get:
\[ \overrightarrow{AB} \times \overrightarrow{AC} = 12\hat{i} - 0\hat{j} - 12\hat{k} = (12, 0, -12) \]

### Step 4: Direction Ratios
The direction ratios of the normal to the plane are \( (12, 0, -12) \).

### Final Answer
The direction ratios of the normal to the plane are \( (12, 0, -12) \).

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