To solve the problem, we need to find the equation of a plane that is perpendicular to the line of intersection of the two given planes \( P_1: x - 2y + 3z = 5 \) and \( P_2: 2x - 3y + z + 4 = 0 \), and passes through the point \( (1, 1, 1) \).
### Step 1: Find the normal vectors of the given planes
The normal vector of a plane given by the equation \( Ax + By + Cz + D = 0 \) is \( (A, B, C) \).
For plane \( P_1: x - 2y + 3z - 5 = 0 \):
- The normal vector \( \mathbf{n_1} = (1, -2, 3) \).
For plane \( P_2: 2x - 3y + z + 4 = 0 \):
- The normal vector \( \mathbf{n_2} = (2, -3, 1) \).
### Step 2: Find the direction ratios of the line of intersection
The direction ratios of the line of intersection of the two planes can be found using the cross product of their normal vectors.
\[
\mathbf{d} = \mathbf{n_1} \times \mathbf{n_2} = (1, -2, 3) \times (2, -3, 1)
\]
Calculating the cross product:
\[
\mathbf{d} = \begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 & -2 & 3 \\
2 & -3 & 1
\end{vmatrix}
\]
Calculating the determinant:
\[
\mathbf{d} = \mathbf{i}((-2) \cdot 1 - 3 \cdot (-3)) - \mathbf{j}(1 \cdot 1 - 3 \cdot 2) + \mathbf{k}(1 \cdot (-3) - (-2) \cdot 2)
\]
\[
= \mathbf{i}(-2 + 9) - \mathbf{j}(1 - 6) + \mathbf{k}(-3 + 4)
\]
\[
= \mathbf{i}(7) - \mathbf{j}(-5) + \mathbf{k}(1)
\]
Thus, the direction ratios of the line of intersection are \( (7, 5, 1) \).
### Step 3: Find the equation of the required plane
The required plane is perpendicular to the line of intersection, which means its normal vector will be the same as the direction ratios of the line of intersection.
Thus, the normal vector of the required plane is \( (7, 5, 1) \).
The general equation of a plane can be written as:
\[
7(x - x_0) + 5(y - y_0) + 1(z - z_0) = 0
\]
Substituting the point \( (1, 1, 1) \):
\[
7(x - 1) + 5(y - 1) + 1(z - 1) = 0
\]
Expanding this:
\[
7x - 7 + 5y - 5 + z - 1 = 0
\]
Combining like terms:
\[
7x + 5y + z - 13 = 0
\]
Thus, the equation of the plane is:
\[
7x + 5y + z = 13
\]
### Final Answer:
The equation of the plane is \( 7x + 5y + z = 13 \).
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