Common Line Indices between the Planes (1bar 1bar 1bar) and (1bar 1 1)

The line indices common to the planes (1bar 1bar 1bar) and (1bar 1 1) can be determined by finding the intersection of the two planes.

Intersection of Two Planes:

To find the intersection of two planes, we need to find the line that lies in both planes. This can be done by finding the direction vector of the common line, which is perpendicular to both planes.

Direction Vector:

The direction vector of a plane can be obtained by taking the cross product of its normal vectors.

Normal Vectors:

The normal vectors of the planes (1bar 1bar 1bar) and (1bar 1 1) can be obtained by looking at the coefficients of the x, y, and z terms in their respective equations.

The equation of the plane (1bar 1bar 1bar) can be written as:

x - y + z = d1

The normal vector of this plane is (1, -1, 1).

Similarly, the equation of the plane (1bar 1 1) can be written as:

x - y - z = d2

The normal vector of this plane is (1, -1, -1).

Cross Product:

Taking the cross product of the normal vectors of the two planes will give us the direction vector of the common line.

(1, -1, 1) x (1, -1, -1) = (-2, 0, 0)

The direction vector of the common line is (-2, 0, 0).

Line Indices:

The line indices of a line can be determined by taking any two points on the line and finding the ratios of their coordinates.

Let's take two points on the common line: (x1, y1, z1) and (x2, y2, z2).

Since the direction vector of the common line is (-2, 0, 0), we can write the following equations:

(x2 - x1)/(-2) = (y2 - y1)/0 = (z2 - z1)/0

Simplifying the equations, we get:

x2 - x1 = 0
y2 - y1 = 0
z2 - z1 = 0

This implies that x2 = x1, y2 = y1, and z2 = z1.

Therefore, the line indices common to the planes (1bar 1bar 1bar) and (1bar 1 1) are:

1. (1bar 1 0)
2. (1 1bar 0)
3. (0 1 1bar)
4. (0 1bar 1)