# Calculus/Lines and Planes in Space

## Contents

## Introduction[edit]

For many practical applications, for example for describing forces in physics and mechanics, you have to work with the mathematical descriptions of **lines** and **planes** in 3-dimensional space.

## Parametric Equations[edit]

### Line in Space[edit]

A line in space is defined by two points in space, which I will call and . Let be the vector from the origin to , and the vector from the origin to . Given these two points, every other point on the line can be reached by

where is the vector from and :

### Plane in Space[edit]

The same idea can be used to describe a plane in 3-dimensional space, which is uniquely defined by three points (which do not lie on a line) in space (). Let be the vectors from the origin to . Then

with:

Note that the starting point does not have to be , but can be any point in the plane. Similarly, the only requirement on the vectors and is that they have to be two non-collinear vectors in our plane.

## Vector Equation (of a *Plane in Space*, or of a *Line in a Plane*)[edit]

An alternative representation of a Plane in Space is obtained by observing that a plane is defined by a point in that plane and a direction perpendicular to the plane, which we denote with the vector .
As above, let describe the vector from the origin to , and the vector from the origin to another point in the plane. Since any vector that lies in the plane is perpendicular to , the **vector equation** of the plane is given by

In 2 dimensions, the same equation uniquely describes a Line.

## Scalar Equation (of a *Plane in Space*, or of a *Line in a Plane*)[edit]

If we express and through their components

writing out the scalar product for
provides us with the **scalar equation** for a plane in space:

where .

In 2d space, the equivalent steps lead to the scalar equation for a line in a plane: