13.1 Positional Relationships of Lines and Planes in Space

13.1 Positional Relationships of Lines and Planes in Space

Solid geometry extends plane synthetic geometry into three-dimensional Euclidean space $\mathbb{R}^3$. To rigorously analyze spatial figures, Gaokao Mathematics establishes a formal axiomatic framework. Understanding positional relationships between spatial lines and planes forms the prerequisite for parallelism proofs, perpendicularity proofs, and spatial vector calculations.

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1. Axiomatic Foundation of Solid Geometry

Euclidean solid geometry rests on four basic axioms that define how points, lines, and planes interact in space:

• Axiom 1 (Plane Determination): Three non-collinear points determine one and only one plane.

  • Symbolic Representation: If $A, B, C$ are not on the same straight line, there exists a unique plane $\alpha$ such that $A, B, C \in \alpha$.
  • Corollaries: A unique plane is also uniquely determined by: (1) A straight line and a point outside it; (2) Two intersecting straight lines; (3) Two parallel straight lines.

• Axiom 2 (Line-Plane Incidence): If two distinct points of a straight line lie in a plane, then the entire straight line lies within that plane.

  • Symbolic Representation: If $A \in l$, $B \in l$, $A \in \alpha$, and $B \in \alpha$ (where $A \neq B$), then $l \subset \alpha$.

• Axiom 3 (Plane Intersection): If two distinct planes share a common point, then they share a unique straight line passing through that point, which is their intersection line.

  • Symbolic Representation: If $P \in \alpha$ and $P \in \beta$ (where $\alpha \neq \beta$), then $\alpha \cap \beta = l$ such that $P \in l$.

• Axiom 4 (Transitivity of Spatial Parallelism): Two straight lines parallel to a third straight line in space are parallel to each other.

  • Symbolic Representation: If $a \parallel b$ and $b \parallel c$, then $a \parallel c$.

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2. Positional Relationships Between Two Lines in Space

Two distinct straight lines $a$ and $b$ in three-dimensional space can be classified according to whether they lie in a common plane and the number of points they share:

Angle Between Two Skew Lines

• Definition: Given two skew lines $a$ and $b$, choose any arbitrary point $O$ in space. Draw line $a' \parallel a$ and line $b' \parallel b$ through $O$. The acute angle or right angle $\theta$ formed by the intersecting lines $a'$ and $b'$ is defined as the angle between the skew lines $a$ and $b$.
• Range: $\theta \in (0^\circ, 90^\circ]$, or in radians $\theta \in \left(0, \frac{\pi}{2}\right]$.
• Perpendicular Skew Lines: If $\theta = 90^\circ$, lines $a$ and $b$ are said to be mutually perpendicular skew lines, denoted $a \perp b$.
• Equal Angles Theorem (等角定理): If two angles in space have their corresponding arms parallel and oriented in the same (or opposite) direction, then the two angles are equal (or supplementary).

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3. Positional Relationships Between a Line and a Plane

A line $a$ and a plane $\alpha$ in space have three mutually exclusive positional relationships:

1. Line lies in the plane ($a \subset \alpha$): Every point on line $a$ lies in plane $\alpha$. There are infinitely many common points.
2. Line is parallel to the plane ($a \parallel \alpha$): Line $a$ and plane $\alpha$ share no common points ($a \cap \alpha = \emptyset$).
3. Line intersects the plane ($a \cap \alpha = {P}$): Line $a$ and plane $\alpha$ share exactly one common point $P$.
   • If line $a$ is perpendicular to every line in plane $\alpha$, $a$ is perpendicular to $\alpha$ ($a \perp \alpha$). Otherwise, $a$ intersects $\alpha$ obliquely.

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4. Positional Relationships Between Two Planes

Two distinct planes $\alpha$ and $\beta$ in space exhibit two fundamental relationships:

1. Parallel Planes ($\alpha \parallel \beta$): The two planes share no common points ($\alpha \cap \beta = \emptyset$).
2. Intersecting Planes ($\alpha \cap \beta = l$): The two planes intersect along a unique straight line $l$. The region bounded by two intersecting half-planes sharing line $l$ is called a dihedral angle (二面角), denoted $\alpha-l-\beta$.

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5. Classic Gaokao Worked Example

Problem

In a standard unit cube $ABCD-A_1B_1C_1D_1$ with edge length $1$:

1. Determine the positional relationship between line $A_1B$ and line $B_1C$.
2. Compute the exact angle between the skew lines $A_1B$ and $B_1C$.

Solution

1. Positional Relationship: Notice that $A_1, B, B_1, C$ are four vertices of the cube. Line $A_1B$ lies in the left face $ABB_1A_1$, while line $B_1C$ lies in the right face $BCC_1B_1$. If $A_1B$ and $B_1C$ were coplanar, all four points $A_1, B, B_1, C$ would lie in one plane. However, $C \notin$ plane $ABB_1A_1$. Furthermore, $A_1B$ and $B_1C$ are not parallel because $\vec{A_1B} = (0, 1, -1)$ and $\vec{B_1C} = (-1, 0, -1)$ in a standard coordinate frame. Thus, $A_1B$ and $B_1C$ are skew lines.

2. Calculating the Angle via Synthetic Translation:

   • Observe that in the face $A_1B_1C_1D_1$, $A_1D_1 \parallel B_1C_1$ and $A_1D_1 = B_1C_1$. In face $ABCD$, $BC \parallel A_1D_1$. In fact, $A_1D \parallel B_1C$.
   • Translate line $B_1C$ to line $A_1D$ (since $B_1C \parallel A_1D$). Both $A_1B$ and $A_1D$ intersect at vertex $A_1$.
   • The angle between skew lines $A_1B$ and $B_1C$ is equal to $\angle BA_1D$ in $\triangle A_1BD$.
   • Connect $BD$. The line segments $A_1B$, $A_1D$, and $BD$ are face diagonals of the cube faces $ABB_1A_1$, $ADD_1A_1$, and $ABCD$, respectively.
   • Each face diagonal of a unit cube has length $\sqrt{1^2 + 1^2} = \sqrt{2}$.
   • Therefore, $\triangle A_1BD$ is an equilateral triangle with side length $\sqrt{2}$.
   • Hence, $\angle BA_1D = 60^\circ = \frac{\pi}{3}$.
   • Conclusion: The angle between the skew lines $A_1B$ and $B_1C$ is $60^\circ$ (or $\frac{\pi}{3}$ radians).