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 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 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 (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