Finding Direction Ratios of Line of Intersection of Planes

The direction ratios of the line of intersection of two planes are determined by the direction vector of that line. This direction vector is perpendicular to the normal vectors of both planes. Therefore, the direction vector can be found by calculating the cross product of the normal vectors of the two planes.

Let's identify the normal vectors for the given planes:

• Plane 1: \( x + y + z = 1 \)
• Plane 2: \( 2x - y - 2z = 3 \)

The general form of a plane equation is \( Ax + By + Cz = D \), where \( \langle A, B, C \rangle \) is the normal vector to the plane.

Calculating the Normal Vectors

• For Plane 1, \( x + y + z = 1 \), the coefficients of \( x, y, \) and \( z \) give the normal vector \( \vec{n_1} = \langle 1, 1, 1 \rangle \).
• For Plane 2, \( 2x - y - 2z = 3 \), the coefficients give the normal vector \( \vec{n_2} = \langle 2, -1, -2 \rangle \).

Calculating the Direction Vector (Cross Product)

The direction vector \( \vec{d} \) of the line of intersection is the cross product of \( \vec{n_1} \) and \( \vec{n_2} \).

We calculate the cross product as follows:

The direction vector is \( \vec{d} = \langle -1, 4, -3 \rangle \). The direction ratios of the line are the components of this vector, i.e., \( \langle -1, 4, -3 \rangle \).

Understanding Direction Ratios and Options

Direction ratios of a line are any set of numbers proportional to the direction vector. If \( \langle a, b, c \rangle \) are direction ratios, then \( \langle ka, kb, kc \rangle \) are also valid direction ratios for any non-zero scalar \( k \).

Our calculated direction ratios are \( \langle -1, 4, -3 \rangle \). We need to find which option is proportional to this vector.

• Option 1: \( \langle 1, 4, 3 \rangle \). Not proportional.
• Option 2: \( \langle -1, -4, 3 \rangle \). Not proportional.
• Option 3: \( \langle 1, -4, 3 \rangle \). Let's check if \( \langle 1, -4, 3 \rangle = k \langle -1, 4, -3 \rangle \). This implies \( 1 = -k \), \( -4 = 4k \), and \( 3 = -3k \). From all three equations, we get \( k = -1 \). Since a consistent non-zero value of \( k \) exists, \( \langle 1, -4, 3 \rangle \) is proportional to \( \langle -1, 4, -3 \rangle \).
• Option 4: \( \langle 1, -4, -3 \rangle \). Not proportional.

Therefore, the direction ratios of the line of intersection of the given planes can be represented by \( \langle 1, -4, 3 \rangle \).