Calculus/Lines and Planes in Space

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Lines and Planes in Space

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 :

Line in 3D Space.

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: