Positions of points, lines and planes in space

1. Use precise incidence conditions; diagrams alone do not prove spatial relationships.

2. Calculate or simplify this relation.

   $$A,B,C\text{ noncollinear }\Rightarrow\exists!\alpha:A,B,C\in\alpha$$

3. Three collinear points instead lie in infinitely many planes.

The requested value is 1.

Checks and common pitfalls: Three collinear points instead lie in infinitely many planes.

1. Use precise incidence conditions; diagrams alone do not prove spatial relationships.

2. Calculate or simplify this relation.

   $$\binom43=4$$

3. If two such planes coincided, all four points would be coplanar.

The requested value is 4.

Checks and common pitfalls: If two such planes coincided, all four points would be coplanar.

1. Use precise incidence conditions; diagrams alone do not prove spatial relationships.

2. Calculate or simplify this relation.

   $$l\parallel m,\ l\ne m\Rightarrow\exists!\alpha:l,m\subseteq\alpha$$

3. The three selected points are noncollinear and determine a unique plane.

The requested value is 1.

Checks and common pitfalls: The three selected points are noncollinear and determine a unique plane.

1. The three-point plane axiom requires noncollinearity.

2. Every plane containing their common line contains all three points; rotating a plane around that line gives infinitely many choices.

3. The missing noncollinearity condition changes uniqueness completely.

Infinitely many.

Checks and common pitfalls: The missing noncollinearity condition changes uniqueness completely.

1. Select two distinct points on the line.

2. Calculate or simplify this relation.

   $$A,B\in l,\ A\ne B,\ P\notin l\Rightarrow A,B,P\text{ noncollinear}$$

3. The unique plane of those three points contains the whole original line.

The requested value is 1.

Checks and common pitfalls: The unique plane of those three points contains the whole original line.

1. Use precise incidence conditions; diagrams alone do not prove spatial relationships.

2. Calculate or simplify this relation.

   $$AB:(t,0,0);\quad CC\prime:(3,3,s)$$

3. Two nonparallel nonintersecting lines in space are skew, not parallel.

Skew lines.

Checks and common pitfalls: Two nonparallel nonintersecting lines in space are skew, not parallel.

1. Use precise incidence conditions; diagrams alone do not prove spatial relationships.

2. Calculate or simplify this relation.

   $$A,B\in\alpha\cap\beta,\ A\ne B\Rightarrow AB\subseteq\alpha\cap\beta$$

3. Distinct intersecting planes meet in a line, not a finite segment.

The entire line through those two points.

Checks and common pitfalls: Distinct intersecting planes meet in a line, not a finite segment.

1. Use precise incidence conditions; diagrams alone do not prove spatial relationships.

2. Calculate or simplify this relation.

   $$Ox\cap Oy\cap Oz=\{O\}$$

3. Pairwise intersections at three distinct points would give a different conclusion.

No: the three Cartesian coordinate axes meet at one point but are not coplanar.

Checks and common pitfalls: Pairwise intersections at three distinct points would give a different conclusion.

1. Use precise incidence conditions; diagrams alone do not prove spatial relationships.

2. Calculate or simplify this relation.

   $$A,B\in\alpha\Rightarrow AB\subseteq\alpha$$

3. One common point permits a line to pass through the plane without lying in it.

The line AB is contained in α by the plane incidence axiom.

Checks and common pitfalls: One common point permits a line to pass through the plane without lying in it.

1. Recall the intersection axiom rather than judging a perspective sketch.

2. Apply the intersection property of planes: nonempty intersection of two distinct planes is an entire straight line.

3. A single common point guarantees many common points.

No; distinct planes that intersect do so in a line.

Checks and common pitfalls: A single common point guarantees many common points.

1. Count unordered triples.

2. Calculate or simplify this relation.

   $$\binom53=5\cdot4\cdot3/(3\cdot2\cdot1)=10$$

3. Both general-position conditions are needed for this count.

The requested value is 10.

Checks and common pitfalls: Both general-position conditions are needed for this count.

1. Take a cross-section perpendicular to the intersection line.

2. Calculate or simplify this relation.

   $$2\times2=4$$

3. The four planar sectors correspond to four spatial regions.

The requested value is 4.

Checks and common pitfalls: The four planar sectors correspond to four spatial regions.

1. Use precise incidence conditions; diagrams alone do not prove spatial relationships.

2. Calculate or simplify this relation.

   $$AB:(t,0,0);\quad CC\prime:(4,4,s)$$

3. Two nonparallel nonintersecting lines in space are skew, not parallel.

Skew lines.

Checks and common pitfalls: Two nonparallel nonintersecting lines in space are skew, not parallel.