Equations of Lines and Planes in Space

Learning Objectives

After studying this chapter, you should be able to:

• Write the vector, parametric, and symmetric equations of a line in 3D space.
• Derive the vector and scalar equations of a plane using a point and a normal vector.
• Determine the relationship between lines and planes (parallel, intersecting, or skew).
• Calculate the distance from a point to a line or a plane.
• Find the line of intersection between two non-parallel planes.

Key Terms & Glossary

• Direction Vector (̅v): A non-zero vector that defines the direction of a line. Example: A car moving along a straight track in a 3D simulation follows a direction vector.
• Normal Vector (̅n): A vector perpendicular to every vector lying within a specific plane. Example: The flagpole is a normal vector to the flat ground of the courtyard.
• Parametric Equations: A set of equations that express the coordinates of the points of a curve as functions of a variable called a parameter ($t$).
• Skew Lines: Two lines in 3D space that are not parallel and do not intersect.
• Symmetric Equations: Equations of a line derived by solving parametric equations for the parameter $t$.

The "Big Idea"

In 2D, a line is often defined by a slope and a point. In 3D, we transition to using vectors. A line is defined by a point and a direction vector (how it stretches), while a plane is defined by a point and a normal vector (what it is perpendicular to). Understanding these relationships allows us to map 3D environments, from architectural designs to computer graphics.

Formula / Concept Box

Hierarchical Outline

• I. Lines in Space
  • A. Requirements: A point $P_0(x_0, y_0, z_0)$ and a direction vector $\mathbf{v} = \langle a, b, c \rangle$.
  • B. Parametric Representation: Describes movement along the line over time $t$.
  • C. Symmetric Representation: Useful for eliminating the parameter $t$.
• II. Planes in Space
  • A. Requirements: A point $P_0(x_0, y_0, z_0)$ and a normal vector $\mathbf{n} = \langle a, b, c \rangle$.
  • B. Determination Methods:
    • Three non-collinear points.
    • Two intersecting lines.
    • A point and a line.
• III. Relative Positions
  • A. Parallel Planes: Normals are parallel (scalar multiples).
  • B. Intersecting Planes: The intersection is always a line.

Visual Anchors

Plane Determination Logic

Geometric Visualization

Definition-Example Pairs

• Term: Scalar Equation of a Plane

• Definition: An algebraic form ax + by + cz + d = 0 where $\langle a, b, c \rangle$ is the normal vector.

• Real-World Example: The surface of a solar panel can be modeled by a scalar equation to determine how sunlight hits the panel at different angles.

• Term: Line of Intersection

• Definition: The set of points shared by two non-parallel planes.

• Real-World Example: The corner where two walls meet in a room is a physical representation of the line of intersection of two planes.

Worked Examples

Example 1: Finding the Equation of a Plane

Problem: Find the equation of the plane passing through the point $P(1, -2, 3)$ with normal vector $\mathbf{n} = \langle 4, 5, -1 \rangle$.

Step-by-Step Breakdown:

1. Identify components: $x_0 = 1, y_0 = -2, z_0 = 3$ and a = 4, b = 5, c = -1.
2. Plug into Scalar Equation: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$.
3. Substitute: 4(x - 1) + 5(y - (-2)) + (-1)(z - 3) = 0.
4. Expand: 4x - 4 + 5y + 10 - z + 3 = 0.
5. Simplify: $4x + 5y - z + 9 = 0$.

[!TIP] Always double-check your normal vector by ensuring it is orthogonal to any vector you can create using points known to be in the plane.

Checkpoint Questions

1. If two planes have normal vectors $\mathbf{n_1} = \langle 1, 2, 3 \rangle$ and $\mathbf{n_2} = \langle -2, -4, -6 \rangle$, are the planes parallel, perpendicular, or neither?
2. What is the minimum information required to define a unique line in 3D space?
3. How do you find a direction vector for the line of intersection between two planes given their normal vectors?
4. Explain why the distance from a point to a plane is defined as the "shortest" distance.

▶Click to see answers

1. Parallel (the normals are scalar multiples: $\mathbf{n_2} = -2\mathbf{n_1}$).
2. A point on the line and a direction vector (or two points on the line).
3. Take the cross product of the two normal vectors: $\mathbf{v} = \mathbf{n_1} \times \mathbf{n_2}$.
4. The shortest distance is the length of the segment perpendicular to the plane from the point.