Let two planes `p _(1): 2x -y + z =2 and p _(2) : x + 2y - z=3` are given :
Equation of the plane which passes through the point `(-1,3,2)` and is perpendicular to each of the plane is:

Options

A. `x + 3y - 5z +2 =0`
B. ` x + 3y + 5z + 18 =0`
C. `x - 3y - 5z + 20=0`
D. `x - 3y + 5z =0`

Answer

To find the equation of the plane that passes through the point \((-1, 3, 2)\) and is perpendicular to the given planes \(P_1: 2x - y + z = 2\) and \(P_2: x + 2y - z = 3\), we can follow these steps:

### Step 1: Identify the normal vectors of the given planes
The normal vector of a plane given by the equation \(Ax + By + Cz = D\) is \((A, B, C)\).

For plane \(P_1: 2x - y + z = 2\), the normal vector \(n_1\) is:
\[ n_1 = (2, -1, 1) \]

For plane \(P_2: x + 2y - z = 3\), the normal vector \(n_2\) is:
\[ n_2 = (1, 2, -1) \]

### Step 2: Find the normal vector of the required plane
Since the required plane is perpendicular to both \(P_1\) and \(P_2\), its normal vector \(n\) can be found by taking the cross product of \(n_1\) and \(n_2\):
\[ n = n_1 \times n_2 \]

Calculating the cross product:
\[ n = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & 1 \\ 1 & 2 & -1 \end{vmatrix} \]

Calculating the determinant:
\[ n = \hat{i}((-1)(-1) - (1)(2)) - \hat{j}((2)(-1) - (1)(1)) + \hat{k}((2)(2) - (-1)(1)) \]
\[ = \hat{i}(1 - 2) - \hat{j}(-2 - 1) + \hat{k}(4 + 1) \]
\[ = -\hat{i} + 3\hat{j} + 5\hat{k} \]
Thus, the normal vector \(n\) is:
\[ n = (-1, 3, 5) \]

### Step 3: Use the point-normal form of the plane equation
The equation of a plane with normal vector \((a, b, c)\) passing through a point \((x_0, y_0, z_0)\) is given by:
\[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \]

Substituting the normal vector \((-1, 3, 5)\) and the point \((-1, 3, 2)\):
\[ -1(x + 1) + 3(y - 3) + 5(z - 2) = 0 \]

### Step 4: Simplify the equation
Expanding the equation:
\[ -x - 1 + 3y - 9 + 5z - 10 = 0 \]
Combining like terms:
\[ -x + 3y + 5z - 20 = 0 \]
Rearranging gives:
\[ x - 3y - 5z + 20 = 0 \]

### Final Equation
The equation of the required plane is:
\[ x - 3y - 5z + 20 = 0 \]

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