Section 11.5 Lines and Planes in Space

Introduction

This section explores the mathematical representation of lines and planes in three-dimensional space using parametric and vector equations. It also covers methods for calculating distances and angles between geometric objects, which are foundational concepts in multivariable calculus and vector analysis.

Lines in Space

Parametric Equations for a Line

A line in space can be described parametrically using a point and a direction vector. The standard parametrization for a line through point P0(x0, y0, z0) parallel to vector v = v1i + v2j + v3k is:

• Parametric equations:

Key Point: Any point on the line can be reached by varying the parameter t.

Example: For a line through (-2, 0, 4) parallel to :

Vector Equation for a Line

The vector equation for a line through P0(x0, y0, z0) parallel to v is:

• r(t): Position vector of a point on the line

• r0: Position vector of the initial point

Parametrizing Line Segments

To parametrize a segment joining points P and Q:

• Use the parametric equations for the line through P and Q

• Restrict the parameter t to

Example: For and :

Distance Calculations

Distance from a Point to a Line

The distance from point S to a line through P parallel to v is given by:

• PS: Vector from P to S

• v: Direction vector of the line

• ×: Cross product

Example: For and line parallel to :

Planes in Space

Equation for a Plane

A plane through P0(x0, y0, z0) normal to n = Ai + Bj + Ck has:

• Vector equation:

• Component equation:

• Simplified: , where

Intersection and Angles Between Planes

The line of intersection of two planes is related to the cross product of their normal vectors:

• Direction vector of intersection:

• Angle between planes: The angle between their normal vectors

Distance from a Point to a Plane

The distance from point S to a plane is the length of the vector projection of onto the normal vector :

Summary Table: Key Equations

Additional info:

• These concepts are essential for understanding vector-valued functions and motion in space (Calculus III topics).

• Applications include physics (motion, force vectors), engineering (design, analysis), and computer graphics (3D modeling).