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Determinants and Area/Volumes of Parallelotopes (Parallelograms and Parallelopipeds) |
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A polytope is a geometric figure with flat sides. Examples of 3 dimensional polytopes (3-polytopes) are a tetrahedron or a parallelopiped. Any polygon is a 2-polytope. The name used for the class of parllelograms and parallelopipeds is parallotopes. A parallogram is a 2-parallelotope and a parallelopiped is a 3-parallelotope.
A parallelopiped (3-parallelotope) is generated from a parallelogram (2-parallelotope) by creating a duplicate of the parallelogram parallel to the original and separated from it by a perpendicular distance d. A parallelogram can be generated from a straight line segment by the same process. A straight line segment can therefore be considered as a 1-parallelotope.
Illustration:
Let A, B and C be three vectors emanating from a corner point of a parallelopiped. One matrix generated from their components is:
|a_{x} | a_{y} | a_{z}| |
|b_{x} | b_{y} | b_{z}| |
|c_{x} | c_{y} | c_{z}| |
The order of the arrangement of the vectors does not matter because the absolute value of the determinant for all orders are the same.
According to the theorem the volume of the associated parallelopiped is the absolute value of the determinant of the above matrix.
Let V_{n} be the generalized volume of an n-parallelotope. Let N be the vector which is normal (perpendicular) in Euclidian (n+1) dimensional space to one face of the n-parallelotope. An (n+1)-parallelotope can be generated from the n-parallelotope by displacing it parallel to itself along a vector A. Let φ denote the acute angle between the vector A and the normal N of the n-parallelotope.
The volume of (n+1)-parallelotope is equal to
Consider a coordinate system which is the coordinate system for the n-parallelotope with the normal N adjoined. The (n+1) dimension component of the vector A is ||A|||cos(φ). The (n+1) dimension components of all of the vectors of the n-parallelotope are zero. The matrix for the (n+1)-parallelotope is then of the form
|a_{1} | … | a_{n} | a_{n+1}| |
|b_{1} | … | b_{n} | 0 | |
… | |||
|h_{1} | … | h_{n} | 0 | |
The absolute value of the determinant of the above matrix is equal to the absolute value of the determinant of this matrix times ||A||cos(φ)
|b_{1} | … | b_{n}| |
… | ||
|h_{1} | … | h_{n}| |
But the absolute value of the determinant of this matrix is simply the generalized volume of the n-parallelotope, V_{n}. But as previously noted
Therefore V_{n+1} is equal to the absolute value of the determinant of the matrix
|a_{1} | … | a_{n} | a_{n+1}| |
|b_{1} | … | b_{n} | 0 | |
… | |||
|h_{1} | … | h_{n} | 0 | |
The quantity
|a_{1} | … | a_{n} | a_{n+1} | |
|b_{1} | … | b_{n} | 0 | |
… | |||
|h_{1} | … | h_{n} | 0 | |
Thus if the theorem is true for an n-parallelotope it is true for an (n+1)-parallelotope.
The final step is to note that the theorem is true for n=1. The extent of a 1-parallelotope is the length of the line segment and the determinant is simply the length of the line segment.
(To be continued.)
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