3.1 Relationships between lines and planes

Lines and Planes in Three-Dimensional Space

In two dimensions, two lines either intersect or they're parallel. Once you move into three-dimensional space, a third possibility appears: lines can be skew, meaning they don't intersect and aren't parallel. This section covers how lines relate to each other and to planes in 3D, which forms the foundation for reasoning about parallel and perpendicular lines throughout the rest of this unit.

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Types of Line Relationships

Parallel lines lie in the same plane and never intersect. They maintain a constant distance between them at all points. Think of railroad tracks extending into the distance.

Skew lines are the relationship that only exists in 3D. Two lines are skew when they don't lie in the same plane and never intersect. Because they exist on different planes, they're neither parallel nor intersecting. A good example: picture the top edge of a door and the bottom edge of the wall across the room. Those two segments aren't parallel, and they never cross.

Intersecting lines share exactly one common point. Two intersecting lines are always coplanar (they define a plane together). Scissors opening and closing illustrate this well.

Types of Line-Plane Relationships

A line and a plane in 3D space relate in exactly one of three ways:

• Parallel to the plane — The line never touches the plane. The distance between them stays constant. Picture a power line running above flat ground.
• Contained in the plane — Every point on the line is also a point on the plane. A pencil lying flat on a desk sits in the plane of the desk.
• Intersecting the plane — The line passes through the plane at exactly one point. A needle pushing through fabric crosses the plane of the fabric at a single point.

A special case of intersection is when the line is perpendicular to the plane, forming a 90° angle at the point of contact (called the foot of the perpendicular). A flagpole standing straight up from level ground is perpendicular to the ground plane.

Classifying Line Positions in 3D

To determine the relationship between two lines in three-dimensional space:

1. Check whether the lines lie in the same plane.

   • If they do, they're either parallel or intersecting.
   • If they don't, they're skew.

2. If the lines are coplanar, check for a common point.

   • No common point → parallel.
   • Exactly one common point → intersecting.

Skew lines fail both tests: they share no common point, yet they aren't parallel because they point in different directions on different planes.

Conditions for Perpendicularity to a Plane

A line being perpendicular to a plane is a stronger condition than it might seem at first. It's not enough for the line to form a 90° angle with one line in the plane. The requirement is:

In practice, you don't need to check every line. A useful theorem simplifies this:

This is why the corner of a room works as an example: the vertical edge where two walls meet is perpendicular to both the front-back direction and the left-right direction along the floor, so it's perpendicular to the floor plane.

Key Properties of Parallel Lines and Planes

• If a line is parallel to a plane, then any line perpendicular to that plane is also perpendicular to the line.
• If two planes are parallel, any line perpendicular to one plane is also perpendicular to the other. Think of a column running between two parallel floors in a building.
• If a line is parallel to a plane, then any plane that contains that line either intersects the original plane or is parallel to it.

Key Properties of Perpendicular Lines and Planes

• If a line is perpendicular to a plane, then any line in the plane passing through the foot is perpendicular to that line.
• If a line is perpendicular to a plane, and a second line is also perpendicular to the same plane, then the two lines are parallel to each other. This is a powerful way to prove lines are parallel in 3D.
• If two planes are perpendicular to each other, a line drawn perpendicular to one plane at a point along their intersection will lie entirely in the other plane. The walls and floor of a room illustrate this: a horizontal line drawn perpendicular to the floor at the base of a wall runs along the wall's surface.