# Calculus/Lines and Planes in Space

 ← Vectors Calculus Multivariable calculus → Lines and Planes in Space

## Introduction

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

### Line in Space

A line in space is defined by two points in space, which I will call $P_{1}$ and $P_{2}$ . Let $\mathbf {x} _{1}$ be the vector from the origin to $P_{1}$ , and $\mathbf {x} _{2}$ the vector from the origin to $P_{2}$ . Given these two points, every other point $P$ on the line can be reached by

$\mathbf {x} =\mathbf {x} _{1}+\lambda \mathbf {a}$ where $\mathbf {a}$ is the vector from $P_{1}$ and $P_{2}$ :

$\mathbf {a} =\mathbf {x} _{2}-\mathbf {x} _{1}$ ### Plane in Space

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 ($P_{1},P_{2},P_{3}$ ). Let $\mathbf {x} _{i}$ be the vectors from the origin to $P_{i}$ . Then

$\mathbf {x} =\mathbf {x} _{1}+\lambda \mathbf {a} +\mu \mathbf {b}$ with:

$\mathbf {a} =\mathbf {x} _{2}-\mathbf {x} _{1}\,\,{\text{and}}\,\,\mathbf {b} =\mathbf {x} _{3}-\mathbf {x} _{1}$ Note that the starting point does not have to be $\mathbf {x} _{1}$ , but can be any point in the plane. Similarly, the only requirement on the vectors $\mathbf {a}$ and $\mathbf {b}$ 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)

An alternative representation of a Plane in Space is obtained by observing that a plane is defined by a point $P_{1}$ in that plane and a direction perpendicular to the plane, which we denote with the vector $\mathbf {n}$ . As above, let $\mathbf {x} _{1}$ describe the vector from the origin to $P_{1}$ , and $\mathbf {x}$ the vector from the origin to another point $P$ in the plane. Since any vector that lies in the plane is perpendicular to $\mathbf {n}$ , the vector equation of the plane is given by

$\mathbf {n} \cdot (\mathbf {x} -\mathbf {x} _{1})=0$ In 2 dimensions, the same equation uniquely describes a Line.

## Scalar Equation (of a Plane in Space, or of a Line in a Plane)

If we express $\mathbf {n}$ and $\mathbf {x}$ through their components

$\mathbf {n} =\left({\begin{array}{*{20}c}a\\b\\c\\\end{array}}\right),\,\,{\text{and}}\,\,\mathbf {x} =\left({\begin{array}{*{20}c}x\\y\\z\\\end{array}}\right),$ writing out the scalar product for $\mathbf {n} \cdot (\mathbf {x} -\mathbf {x} _{1})=0$ provides us with the scalar equation for a plane in space:

$ax+by+cz=d$ where $d=\mathbf {n} \cdot \mathbf {x} _{1}$ .

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

$ax+by=c$ 