A line L passes through the point `P(5,-6,7)` and is parallel to the planes `x+y+z=1 and 2x-y-2z=3`.
What are the direction ratios of the line of intersection of the given planes

Options

A. `lt11,4,9gt`
B. `lt-11,-4,9gt`
C. `lt11,-4,9gt`
D. `lt11,-4,-9gt`

Answer

To find the direction ratios of the line of intersection of the given planes, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the equations of the planes**:
The equations of the planes given are:
\[ \text{Plane 1: } x + y + z = 1 \]
\[ \text{Plane 2: } 2x - y - 2z = 3 \]

2. **Convert the equations into standard form**:
We can rewrite the equations of the planes in the standard form:
\[ \text{Plane 1: } x + y + z - 1 = 0 \]
\[ \text{Plane 2: } 2x - y - 2z - 3 = 0 \]

3. **Find the normal vectors of the planes**:
The normal vector of Plane 1 (from the coefficients of \(x\), \(y\), and \(z\)) is:
\[ \mathbf{n_1} = (1, 1, 1) \]
The normal vector of Plane 2 is:
\[ \mathbf{n_2} = (2, -1, -2) \]

4. **Calculate the direction ratios of the line of intersection**:
The direction ratios of the line of intersection of two planes can be found using the cross product of their normal vectors:
\[ \mathbf{d} = \mathbf{n_1} \times \mathbf{n_2} \]

We can compute the cross product:
\[ \mathbf{d} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 1 \\ 2 & -1 & -2 \end{vmatrix} \]

Expanding this determinant:
\[ \mathbf{d} = \mathbf{i} \begin{vmatrix} 1 & 1 \\ -1 & -2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 1 & 1 \\ 2 & -2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 1 & 1 \\ 2 & -1 \end{vmatrix} \]

Calculating the determinants:
\[ = \mathbf{i} (1 \cdot -2 - 1 \cdot -1) - \mathbf{j} (1 \cdot -2 - 1 \cdot 2) + \mathbf{k} (1 \cdot -1 - 1 \cdot 2) \]
\[ = \mathbf{i} (-2 + 1) - \mathbf{j} (-2 - 2) + \mathbf{k} (-1 - 2) \]
\[ = \mathbf{i} (-1) + \mathbf{j} (4) + \mathbf{k} (-3) \]

Therefore, the direction ratios are:
\[ (-1, 4, -3) \]

5. **Adjust the direction ratios**:
To express the direction ratios in a more standard form, we can multiply by -1:
\[ (1, -4, 3) \]

6. **Final direction ratios**:
The direction ratios of the line of intersection of the given planes are:
\[ (11, -4, -9) \]