Lines and planes in space

Space holds points, lines and planes. The box $\displaystyle ABCDA'B'C'D'$ is a handy model of all three:

• Each vertex is a point, and each edge is part of a line. The line goes on past the edge in both directions.
• Each face is part of a plane. The plane goes on past the face in every direction, like a floor with no walls.
• A plane is named by three or more of its points: the base plane $\displaystyle ABCD$, the plane $\displaystyle ABB'$.
• In a drawing, an edge hidden behind the solid is drawn dashed.

Through three points that are not on one line there passes exactly one plane. So a plane is determined by any of these:

• three points not on one line – for example $\displaystyle A$, $\displaystyle B$, $\displaystyle C$ determine the base plane;
• a line and a point not on it – the line $\displaystyle AB$ and the point $\displaystyle D$;
• two intersecting lines – $\displaystyle AB$ and $\displaystyle BC$, which meet at $\displaystyle B$;
• two parallel lines – $\displaystyle AB$ and $\displaystyle DC$.

Each of the last three hides three points: two points on one line, and one more off it.

Two different lines in space are in one of three positions:

• intersecting – they have one common point. They lie in one plane.
• parallel – they lie in one plane and have no common point.
• skew – no plane holds them both. So they never meet, and they are not parallel either.

In the drawing, $\displaystyle AB$ and $\displaystyle CC'$ are skew: $\displaystyle AB$ lies in the base, and $\displaystyle CC'$ leaves the base upwards at $\displaystyle C$, which is not on $\displaystyle AB$.

In a plane there are no skew lines: two lines in a plane that do not meet are always parallel.

In the box $\displaystyle ABCDA'B'C'D'$:

• $\displaystyle AB$ and $\displaystyle A'B'$ are parallel: the face $\displaystyle ABB'A'$ is a rectangle.
• $\displaystyle AB$ and $\displaystyle BB'$ intersect at $\displaystyle B$.
• $\displaystyle AC$ and $\displaystyle B'D'$ are skew. $\displaystyle AC$ lies in the base plane, and $\displaystyle B'D'$ in the top’s plane. These planes never meet, so the lines never meet. Nor are they parallel: $\displaystyle B'D'$ is parallel to $\displaystyle BD$, and $\displaystyle BD$ crosses $\displaystyle AC$.

In the box $\displaystyle ABCDA'B'C'D'$, what is the position of each pair of lines? Explain.

$\displaystyle AA'$ and $\displaystyle CC'$

$\displaystyle A'C'$ and $\displaystyle CC'$

$\displaystyle AB$ and $\displaystyle DD'$

$\displaystyle BD$ and $\displaystyle B'D'$

$\displaystyle SABCD$ is a pyramid whose base is the parallelogram $\displaystyle ABCD$.
Write what determines the base plane:

a point and a line

two intersecting lines

two parallel lines

In the same pyramid $\displaystyle SABCD$, what is the position of each pair of lines? Explain.

$\displaystyle DC$ and $\displaystyle AB$

$\displaystyle DC$ and $\displaystyle SB$

$\displaystyle SA$ and $\displaystyle SC$

A three-legged stool stands firm on any floor, but a four-legged chair sometimes wobbles. Explain why, using how a plane is determined.