Spatial Geometry: Points, Lines & Planes in 3D

Why Study Spatial Geometry?

Spatial geometry extends 2D reasoning to three dimensions. It develops spatial intuition that is essential for engineering, architecture, physics and computer graphics. The key challenge is reasoning about objects you cannot easily draw on paper.

Learning Goals: By the end of this guide, you should be able to:

Identify the relationships between points, lines and planes (parallel, intersecting, skew).

Calculate distances between points, lines and planes in 3D.

Find angles between lines, between a line and a plane, and dihedral angles.

Solve problems involving common 3D solids (cuboids, pyramids, prisms).

Key Relationships in 3D

Line-Line Relationships

Relationship	Definition
Parallel	Lines never meet and point in the same direction
Intersecting	Lines cross at a single point
Skew	Lines do not intersect AND are not parallel (only in 3D)

Line-Plane Relationships

Relationship	Definition
Line lies in the plane	Every point of the line is on the plane
Line is parallel to the plane	Line does not intersect the plane
Line intersects the plane	They meet at exactly one point
Line is perpendicular to the plane	Line is perpendicular to every line in the plane through the foot

Plane-Plane Relationships

Two distinct planes are either parallel or they intersect along a line.

Distance Formulas

Distance Between Two Points

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}

Distance from a Point to a Plane

For the plane $ax + by + cz + d = 0$ and point $(x_0, y_0, z_0)$ :

D = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}}

Angle Calculations

Angle Between Two Lines

If the direction vectors are $\vec{u}$ and $\vec{v}$ :

\cos\theta = \frac{|\vec{u} \cdot \vec{v}|}{|\vec{u}||\vec{v}|}

Angle Between a Line and a Plane

If the line has direction $\vec{u}$ and the plane has normal $\vec{n}$ :

\sin\alpha = \frac{|\vec{u} \cdot \vec{n}|}{|\vec{u}||\vec{n}|}

Dihedral Angle

The dihedral angle between two planes is the angle between their normal vectors (or its supplement).

Worked Example

Question: In a cube $ABCDA'B'C'D'$ with side length 1, find the angle between the diagonal $AG$ (from vertex $A$ to opposite vertex $G$ ) and the base plane $ABCD$ .

Setup: Place $A$ at origin. Then $A = (0,0,0)$ and $G = (1,1,1)$ .

Direction of $AG$ : $\vec{u} = (1, 1, 1)$

Normal to base plane: $\vec{n} = (0, 0, 1)$

\sin\alpha = \frac{|(1)(0) + (1)(0) + (1)(1)|}{\sqrt{3} \cdot 1} = \frac{1}{\sqrt{3}}

\alpha = \arcsin\frac{1}{\sqrt{3}} \approx 35.3°

Common Mistakes

Assuming all non-intersecting lines are parallel — In 3D, lines can be skew: non-intersecting and non-parallel. This has no analogue in 2D.
Using the wrong angle formula — For the angle between a line and a plane, use $\sin\alpha$ (not $\cos\alpha$ ). The angle is the complement of the angle with the normal.
Not setting up coordinates — Many 3D problems become straightforward once you place the solid on a coordinate system.

Exam Tips (A-Level / AP / IB)

For cube and cuboid problems, always set up coordinates with one vertex at the origin.
Draw clear diagrams and label all vertices. Mark the angle you are finding.
The foot of a perpendicular from a point to a plane is found by projecting along the normal vector.

Frequently Asked Questions

What are skew lines?

Skew lines are lines in 3D that are neither parallel nor intersecting. Imagine one line along the floor and another along a wall — they don't meet and point in different directions. This is impossible in 2D.

How do I find the equation of a plane?

A plane can be defined by a point $(x_0, y_0, z_0)$ and a normal vector $(a, b, c)$ : $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$ .

Vector Operations — Vectors provide the algebraic tools for 3D geometry calculations.
Conic Sections — Cross-sections of 3D solids produce conic curves.
Trigonometric Graphs — Trigonometry is essential for angle calculations in 3D.

Try the interactive simulation for this topic