A plane P passes through the line of intersection of the planes `2x-y+3z=2.x+y-z=1` and the point `(1,0,1)`
What are the direction ration of the line of intersection of the given planes

Options

A. `lt2,-5,-3gt`
B. `lt1,-5,-3gt`
C. `lt2,5,3gt`
D. `lt1,3,5gt`

Answer

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

### Step 1: Identify the normal vectors of the planes
The equations of the planes are:
1. Plane 1: \(2x - y + 3z = 2\)
2. Plane 2: \(x + y - z = 1\)

From these equations, we can extract the normal vectors:
- For Plane 1, the normal vector \( \mathbf{n_1} = (2, -1, 3) \)
- For Plane 2, the normal vector \( \mathbf{n_2} = (1, 1, -1) \)

### Step 2: 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.

The cross product \( \mathbf{n_1} \times \mathbf{n_2} \) can be calculated as follows:

\[ \mathbf{n_1} \times \mathbf{n_2} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -1 & 3 \\ 1 & 1 & -1 \end{vmatrix} \]

### Step 3: Compute the determinant
Calculating the determinant, we have:

\[ \mathbf{n_1} \times \mathbf{n_2} = \mathbf{i} \begin{vmatrix} -1 & 3 \\ 1 & -1 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 2 & -1 \\ 1 & 1 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} -1 & 3 \\ 1 & -1 \end{vmatrix} = (-1)(-1) - (3)(1) = 1 - 3 = -2 \)
2. \( \begin{vmatrix} 2 & 3 \\ 1 & -1 \end{vmatrix} = (2)(-1) - (3)(1) = -2 - 3 = -5 \)
3. \( \begin{vmatrix} 2 & -1 \\ 1 & 1 \end{vmatrix} = (2)(1) - (-1)(1) = 2 + 1 = 3 \)

Putting it all together:

\[ \mathbf{n_1} \times \mathbf{n_2} = -2 \mathbf{i} + 5 \mathbf{j} + 3 \mathbf{k} \]

Thus, the direction ratios of the line of intersection are \( (-2, 5, 3) \).

### Step 4: Write the final answer
The direction ratios of the line of intersection of the given planes are \( (-2, 5, 3) \).