# Calculus/Lines and Planes in Space

 ← Vectors Calculus Multivariable calculus → Lines and Planes in Space

## IntroductionEdit

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 EquationsEdit

### Line in SpaceEdit

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

${\displaystyle \mathbf {x} =\mathbf {x} _{1}+\lambda \mathbf {a} }$

where ${\displaystyle \mathbf {a} }$  is the vector from ${\displaystyle P_{1}}$  and ${\displaystyle P_{2}}$ :

${\displaystyle \mathbf {a} =\mathbf {x} _{2}-\mathbf {x} _{1}}$

### Plane in SpaceEdit

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

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

with:

${\displaystyle \mathbf {a} =\mathbf {x} _{2}-\mathbf {x} _{1}\,\,{\text{and}}\,\,\mathbf {b} =\mathbf {x} _{3}-\mathbf {x} _{1}}$

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

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

${\displaystyle \mathbf {n} \cdot (\mathbf {x} -\mathbf {x} _{1})=0}$

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

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

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

${\displaystyle \mathbf {n} =\left({\begin{array}{*{20}c}a\\b\\c\\\end{array}}\right),\,\,{\text{and}}\,\,\mathbf {x} =\left({\begin{array}{*{20}c}x\\y\\z\\\end{array}}\right),}$

writing out the scalar product for ${\displaystyle \mathbf {n} \cdot (\mathbf {x} -\mathbf {x} _{1})=0}$  provides us with the scalar equation for a plane in space:

${\displaystyle ax+by+cz=d}$

where ${\displaystyle d=\mathbf {n} \cdot \mathbf {x} _{1}}$ .

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

${\displaystyle ax+by=c}$