Finding Direction Ratios of a Line from Plane Intersection

The problem asks for the direction ratios of a line defined by the intersection of two planes. A line lying in the intersection of two planes is perpendicular to the normal vectors of both planes. Therefore, the direction ratios of the line can be found by calculating the cross product of the normal vectors of the given planes.

Given Plane Equations

The line L is defined by the intersection of the following two planes:

• Plane 1: \(x + y + z + 4 = 0\)
• Plane 2: \(2x - y - z + 8 = 0\)

Note: The plane P (\(x+2y+3z + 1 = 0\)) is extra information and not needed to find the direction ratios of line L.

Normal Vectors

The normal vector to a plane \(Ax + By + Cz + D = 0\) is given by the coefficients \((A, B, C)\).

• The normal vector for Plane 1 (\(N_1\)) is \((1, 1, 1)\).
• The normal vector for Plane 2 (\(N_2\)) is \((2, -1, -1)\).

Calculating Direction Ratios using Cross Product

The direction ratios of the line of intersection are given by the cross product \(N_1 \times N_2\). We can calculate this using the determinant formula:

\( N_1 \times N_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 1 \\ 2 & -1 & -1 \end{vmatrix} \)

Expanding the determinant:

\( N_1 \times N_2 = \mathbf{i} \left( (1)(-1) - (1)(-1) \right) - \mathbf{j} \left( (1)(-1) - (1)(2) \right) + \mathbf{k} \left( (1)(-1) - (1)(2) \right) \)

\( N_1 \times N_2 = \mathbf{i} \left( -1 - (-1) \right) - \mathbf{j} \left( -1 - 2 \right) + \mathbf{k} \left( -1 - 2 \right) \)

\( N_1 \times N_2 = \mathbf{i} \left( -1 + 1 \right) - \mathbf{j} \left( -3 \right) + \mathbf{k} \left( -3 \right) \)

\( N_1 \times N_2 = 0\mathbf{i} + 3\mathbf{j} - 3\mathbf{k} \)

So, the direction ratios obtained from the cross product are \((0, 3, -3)\).

Interpreting Direction Ratios

Direction ratios are not unique; any non-zero multiple of a set of direction ratios also represents the direction of the line. The calculated direction ratios are \((0, 3, -3)\). We can simplify these by dividing each component by a common factor, such as 3:

• \(\frac{0}{3} = 0\)
• \(\frac{3}{3} = 1\)
• \(\frac{-3}{3} = -1\)

This gives us the simplified direction ratios as \((0, 1, -1)\).

Comparing with Options

Comparing our result \((0, 1, -1)\) with the given options:

• Option 1: \((2, 1, -1)\)
• Option 2: \((0, -1, 2)\)
• Option 3: \((0, 1, -1)\)
• Option 4: \((2, 3, -3)\)

Our calculated direction ratios \((0, 1, -1)\) match Option 3.