Equation of the plane passing through the points `i+j-2k,2i-j+kandi+2j+k` is

Options

A. `r.(4i+2j)=20`
B. `r.(9i+3j-k)=14`
C. `r.(9i+3j-k)=6`
D. none of these

Answer

To find the equation of the plane passing through the points \( A(1, 1, -2) \), \( B(2, -1, 1) \), and \( C(1, 2, 1) \), we can follow these steps:

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

\[ \vec{AB} = B - A = (2 - 1, -1 - 1, 1 - (-2)) = (1, -2, 3) \]

\[ \vec{AC} = C - A = (1 - 1, 2 - 1, 1 - (-2)) = (0, 1, 3) \]

### Step 2: Find the normal vector to the plane
Next, we find the normal vector \( \vec{n} \) to the plane by taking the cross product of \( \vec{AB} \) and \( \vec{AC} \).

\[ \vec{n} = \vec{AB} \times \vec{AC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & 3 \\ 0 & 1 & 3 \end{vmatrix} \]

Calculating the determinant:

\[ \vec{n} = \hat{i} \begin{vmatrix} -2 & 3 \\ 1 & 3 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 3 \\ 0 & 3 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & -2 \\ 0 & 1 \end{vmatrix} \]

Calculating each of these determinants:

1. \( \begin{vmatrix} -2 & 3 \\ 1 & 3 \end{vmatrix} = (-2)(3) - (3)(1) = -6 - 3 = -9 \)
2. \( \begin{vmatrix} 1 & 3 \\ 0 & 3 \end{vmatrix} = (1)(3) - (3)(0) = 3 \)
3. \( \begin{vmatrix} 1 & -2 \\ 0 & 1 \end{vmatrix} = (1)(1) - (-2)(0) = 1 \)

Putting it all together:

\[ \vec{n} = -9\hat{i} - 3\hat{j} + 1\hat{k} = (-9, -3, 1) \]

### Step 3: Use the normal vector and a point to find the equation of the plane
The equation of the plane can be expressed as:

\[ \vec{n} \cdot (\vec{r} - \vec{a}) = 0 \]

Where \( \vec{a} \) is a point on the plane (we can use point \( A(1, 1, -2) \)) and \( \vec{r} = (x, y, z) \).

Substituting:

\[ (-9, -3, 1) \cdot ((x, y, z) - (1, 1, -2)) = 0 \]

This expands to:

\[ -9(x - 1) - 3(y - 1) + 1(z + 2) = 0 \]

Simplifying this:

\[ -9x + 9 - 3y + 3 + z + 2 = 0 \]

Combining like terms:

\[ -9x - 3y + z + 14 = 0 \]

Rearranging gives us the equation of the plane:

\[ 9x + 3y - z = 14 \]

### Final Answer
The equation of the plane passing through the points \( i + j - 2k \), \( 2i - j + k \), and \( i + 2j + k \) is:

\[ 9x + 3y - z = 14 \]