Last modified on 20 June 2011, at 21:29

Calculus/Lines and Planes in Space

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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 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

Line in 3D Space.

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 (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)Edit

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)Edit

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