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Unformatted text preview: Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to be contained in some ambient Euclidean space, but each will have its own innate dimension. We shall usually denote the ambient space as R N , and we use its usual inner product • and norm. In discussing these parallelograms, we shall always for simplicity assume one vertex is the origin 0 ∈ R N . This restriction is of course easily overcome in applications by a simple translation of R N . DEFINITION. Let x 1 ,...,x n be arbitrary points in R N (notice that these subscripts do not here indicate coordinates). We then form the set of all linear combinations of these points of a certain kind: P = ( n X i =1 t i x i | ≤ t i ≤ 1 ) . We call this set an n-dimensional parallelogram (with one “vertex” 0). We also refer to the vectors x 1 ,...,x n as the edges of P . • For n = 1 this “parallelogram” is of course just the line segment [0 ,x 1 ]. 1 x O • For n = 2 we obtain a “true” parallelogram. x 1 x 2 O • For n = 3 the word usually employed is parallelepiped . x 2 x 1 x 3 O However, it seems to be a good idea to pick one word to be used for all n , so we actually call this a 3-dimensional “parallelogram.” We are not assuming that the edge vectors x 1 ,...,x n are linearly independent, so it could happen that a 3-dimensional parallelogram could lie in a plane and itself be a 2-dimensional parallelogram, for instance. PROBLEM 8–1. Consider the “parallelogram” in R 3 with “edges” equal to the three points ˆ ı , ˆ , ˆ ı- 2ˆ . Draw a sketch of it and conclude that it is actually a six-sided figure in the x- y plane. A. Volumes in dimensions 1, 2, 3 2 Chapter 8 We are going to work out a definition of the n-dimensional volume of an n-dimensional parallelogram P , and we shall denote this as vol( P ); if we wish to designate the dimension, we write vol n ( P ). In this section we are going to recall some formulas we have already obtained, and cast them in a somewhat different format. • Dimension 1 . We have the line segment [0 ,x 1 ] ⊂ R N , and we simply say its 1-dimensional volume is its length , i.e. the norm of x 1 : vol([0 ,x 1 ]) = k x 1 k . • Dimension 3 . We skip ahead to this dimension because of our already having obtained the pertinent formula in Section 7C. Namely, suppose P ⊂ R 3 , so we are dealing at first with the dimension of P equal to the dimension of the ambient space. Then if we write x 1 , x 2 , x 3 as column vectors, vol( P ) = | det( x 1 x 2 x 3 ) | , the absolute value of the determinant of the 3 × 3 matrix formed from the column vectors....
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