To determine which line segments among AD, AB, AC, and BC intersect the plane given by the equation \(2x - y - 3z = 5\), we will follow these steps:
### Step 1: Identify the points and their coordinates
- Point A: \( (1, 1, 1) \)
- Point B: \( (2, 1, -3) \)
- Point C: \( (1, -2, -2) \)
- Point D: \( (-3, 1, 2) \)
### Step 2: Substitute the coordinates of each point into the plane equation
We will substitute the coordinates of each point into the equation of the plane to determine which side of the plane each point lies on.
1. **For Point A (1, 1, 1)**:
\[
2(1) - 1 - 3(1) = 2 - 1 - 3 = -2
\]
Since \(-2 < 5\), Point A lies on the side of the plane where \(2x - y - 3z < 5\).
2. **For Point B (2, 1, -3)**:
\[
2(2) - 1 - 3(-3) = 4 - 1 + 9 = 12
\]
Since \(12 > 5\), Point B lies on the side of the plane where \(2x - y - 3z > 5\).
3. **For Point C (1, -2, -2)**:
\[
2(1) - (-2) - 3(-2) = 2 + 2 + 6 = 10
\]
Since \(10 > 5\), Point C lies on the same side of the plane as Point B.
4. **For Point D (-3, 1, 2)**:
\[
2(-3) - 1 - 3(2) = -6 - 1 - 6 = -13
\]
Since \(-13 < 5\), Point D lies on the same side of the plane as Point A.
### Step 3: Determine which line segments intersect the plane
- **Line Segment AD**: Points A and D are on opposite sides of the plane (A is less than 5, D is less than 5). Thus, line segment AD intersects the plane.
- **Line Segment AB**: Points A and B are on opposite sides of the plane (A is less than 5, B is greater than 5). Thus, line segment AB intersects the plane.
- **Line Segment AC**: Points A and C are on opposite sides of the plane (A is less than 5, C is greater than 5). Thus, line segment AC intersects the plane.
- **Line Segment BC**: Points B and C are on the same side of the plane (both are greater than 5). Thus, line segment BC does not intersect the plane.
### Conclusion
The line segments that intersect the plane are:
- Line Segment AB
- Line Segment AC
- Line Segment AD
### Final Answer
The line segments that intersect the plane are: **AB, AC, and AD**.
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