Lines and planes in 3D space have fascinating relationships. They can be parallel, skew, or intersecting, each with unique properties. Understanding these connections helps visualize complex structures and solve real-world problems.

From railroad tracks to helicopter blades, these concepts appear everywhere. Recognizing line-plane orientations is crucial for architecture, engineering, and even everyday tasks like hanging a picture straight on a wall.

Lines and Planes in Three-Dimensional Space

Types of line-plane relationships

• Parallel lines
  • Lie in the same plane without intersecting (railroad tracks)
  • Maintain a constant distance between them at all points
• Skew lines
  • Do not lie in the same plane and never intersect (helicopter blades)
  • Non-parallel and non-intersecting in 3D space
• Intersecting lines
  • Share a common point of intersection (scissors)
  • Can be coplanar (in the same plane) or non-coplanar (in different planes)
• Line parallel to a plane
  • Does not intersect the plane at any point (ceiling and floor)
  • Maintains a constant distance from the plane in all directions
• Line perpendicular to a plane
  • Forms a 90° angle with the plane at the point of intersection (flagpole and ground)
  • Intersects the plane at a single point called the foot of the perpendicular
• Line intersecting a plane
  • Passes through the plane at a single point (needle and fabric)
  • Forms angles with the plane that are not 90° (acute or obtuse)

Classification of line positions

• Determine the relationship between two lines in 3D space by:
  1. Checking if the lines lie in the same plane
     • If they do, they are either parallel or intersecting
     • If they do not, they are skew (like a jungle gym)
  2. For lines in the same plane, checking for common points
     • No common points indicate parallel lines
     • A common point indicates intersecting lines (crossroads)
• Skew lines never have a common point and are not parallel (like telephone wires)

Conditions for line-plane orientations

• A line is parallel to a plane when:
  • The line does not intersect the plane at any point (like a balcony and the ground)
  • The line and plane do not share any common points
  • The distance between the line and plane remains constant (like power lines and the earth)
• A line is perpendicular to a plane when:
  • The line forms a 90° angle with the plane (like a lamp post and the sidewalk)
  • The line intersects the plane at a single point, the foot of the perpendicular
  • The line is perpendicular to every line in the plane passing through the foot (like a tree trunk and the ground)

Applications of line-plane properties

• Properties of parallel lines and planes:
  • A line parallel to a plane $\implies$ any line $\perp$ to the plane is also $\perp$ to the line
  • Two parallel planes $\implies$ any line $\perp$ to one plane is also $\perp$ to the other
  • A line parallel to a plane $\implies$ any plane containing the line is parallel to the original plane (like sheets of paper in a stack)
• Properties of perpendicular lines and planes:
  • A line $\perp$ to a plane $\implies$ any line $\perp$ to the line at the intersection point is contained in the plane
  • A line $\perp$ to two intersecting lines in a plane $\implies$ the line is $\perp$ to the plane (like the corner of a room)
  • Two $\perp$ planes $\implies$ any line $\perp$ to one plane at the intersection is contained in the other plane (like the walls and floor of a room)