Calculus/Lines and Planes in Space

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Lines and Planes in Space

Contents

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   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 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 ( ). 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: