Question

Two distinct lines in space can intersect, be ________ or be ________.

Short answer

Parallel and Skew.

Step-by-step solution

Understanding Line Relationships

Two distinct lines in space can exhibit one of three relationships: they can either intersect, be parallel, or be skew. Skew lines are a concept specific to three-dimensional space and indicate lines that do not intersect at any point and are not parallel to each other.

Intersecting Lines

When two lines intersect, they have exactly one point in common. This means there is a point where both lines cross each other. The angle at which they intersect can vary, and if the lines are neither parallel nor skew, they will intersect.

Parallel Lines

Parallel lines are two lines that never intersect. These lines are always the same distance apart and run in the same direction. In space, parallel lines lie on the same plane and extend infinitely without meeting each other.

Skew Lines

Skew lines are lines that do not intersect and are not parallel. This condition is unique to three-dimensional space because skew lines do not lie on the same plane. They can be thought of as 'misaligned' from each other, neither crossing nor running in the same direction.

Key concepts

Intersecting Lines

When discussing lines in space, intersecting lines are one of the primary relationships. Simply put, two lines intersect when they cross each other at a specific point. This single point of intersection is the only place where both lines meet.

• Intersecting lines always meet at some angle, except when they are perpendicular, which means they meet at a 90-degree angle.
• Once they intersect, their path diverges and continues in different directions.

Think of intersecting lines as roads that cross one another at a junction, where each road maintains its unique path but meets at one distinct spot.

Parallel Lines

Parallel lines are another important concept when visualizing line relationships in space. These are pairs of lines that never meet, no matter how far they are extended.

• They maintain a constant distance between them, which means they do not converge or diverge from each other.
• Parallel lines always lie in the same plane and run in a similar direction.

Imagine a set of railway tracks. They are parallel because they run side by side and never come closer or farther apart. This makes them perfectly equidistant at every point.

Skew Lines

Skew lines introduce a fascinating twist to how lines can relate to each other, as they do not fit into the simple categories of intersecting or parallel. These lines exist only in three-dimensional space, which adds to their unique nature.

• Skew lines are neither intersecting nor parallel, making them appear "misaligned" when visualized in 3D space.
• They do not lie in the same plane, which differentiates them significantly from parallel lines.

Imagine two escalators in a shopping mall that go in different directions and have no chance of meeting or running parallel. This concept is particularly useful in architecture and engineering when considering the placement and alignment of various structures.