Direction Ratios of Line Intersection: Step-by-Step Calculation

This question asks for the direction ratios of the line formed by the intersection of two specific planes. Let's break down how to find this.

Understanding the Problem

We are given two planes:

• Plane 1: \(6x+4y-5z = 2\)
• Plane 2: \(x-2y+3z = 0\)

We need to find the direction ratios of the line where these two planes intersect. The direction ratios represent the components of a vector parallel to the line.

Key Concepts

• Normal Vector: For a plane defined by the equation \(Ax + By + Cz = D\), the vector \(\vec{n} = (A, B, C)\) is perpendicular (normal) to the plane.
• Line of Intersection: The line of intersection of two non-parallel planes is perpendicular to the normal vectors of both planes.
• Cross Product: The cross product of two vectors yields a third vector that is perpendicular to both original vectors. If \(\vec{a} = (a_1, a_2, a_3)\) and \(\vec{b} = (b_1, b_2, b_3)\), their cross product is \(\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)\).

Calculating Direction Ratios of Line Intersection

To find the direction ratios of the line of intersection, we can take the cross product of the normal vectors of the two given planes.

The normal vector for the first plane (\(6x+4y-5z = 2\)) is \(\vec{n_1} = (6, 4, -5)\).

The normal vector for the second plane (\(x-2y+3z = 0\)) is \(\vec{n_2} = (1, -2, 3)\).

Step-by-Step Cross Product

Let the direction ratios of the line of intersection be \((a, b, c)\). This vector \((a, b, c)\) is parallel to \(\vec{n_1} \times \vec{n_2}\).

Calculate the cross product:

$\vec{n_1} \times \vec{n_2} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 6 & 4 & -5 \\ 1 & -2 & 3 \end{vmatrix}$

Expanding the determinant:

\(= \mathbf{i} \left( (4)(3) - (-5)(-2) \right) - \mathbf{j} \left( (6)(3) - (-5)(1) \right) + \mathbf{k} \left( (6)(-2) - (4)(1) \right)\)

\(= \mathbf{i} (12 - 10) - \mathbf{j} (18 - (-5)) + \mathbf{k} (-12 - 4)\)

\(= \mathbf{i} (2) - \mathbf{j} (18 + 5) + \mathbf{k} (-16)\)

\(= 2\mathbf{i} - 23\mathbf{j} - 16\mathbf{k}\)

Therefore, the direction ratios of the line of intersection are \((2, -23, -16)\).

Comparing with Options

Let's compare our calculated direction ratios with the given options:

• 1. (2, 23, 16)
• 2. (2, -23, -16)
• 3. (2, 3, 2)
• 4. (-1, 3, -2)

Our result \((2, -23, -16)\) matches the second option.