15.2 Representing Parallel Sided Figures

The first thing we need is a way to describe parallel sided figures.

And here is one way. Imagine we have x and y coordinates in the plane, and we put one corner of our figure at the origin, by which we mean the point \((0, 0)\), for which \(x = 0, \, y = 0\).

Then suppose that the "corners" of the parallelogram that are at the other end of edges that contain the origin, are located at points \((a, b)\) and \((c, d)\). The last corner will be at \((a + c, b + d)\) because the sides are parallel. (Choose values for \(a, b, c\) and \(d\) and draw yourself a figure to verify this statement.)

Then, one way to describe the parallelogram is to give a square array consisting of the numbers \(a, b, c\), and \(d\), arranged as follows:

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

(The rows here are the vectors defined by the edges of the parallelogram that meet the origin.)

For example, the parallelogram with corners \((0, 0), (1, 2), (0, 1)\) and \((1, 3)\) can be represented by the array

\[ \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \]

And in three dimensions, we can describe a parallelepiped with one corner at the origin, \((0, 0, 0)\), by putting the coordinates of the three corners that share edges with the origin as the three rows of a \(3\) by \(3\) array.

And in one dimension we can represent a line segment, which starts at the origin and goes to a point \(x\), by the single entry, \(x\).

Thus, the three matrices

\[ \begin{pmatrix} 5 \end{pmatrix}, \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 8 \\ 1 & 4 & 1 \end{pmatrix} \]

can represent a line segment, a parallelogram and a parallelepiped respectively.

We give a uniform name to the (signed) length of the line segment, the area of the parallelogram and the volume of the parallelepiped, all with some appropriate sign.

Each is called the determinant of the given array. And we can define the determinant just as well with similar meaning for larger square arrays of numbers.

The determinant of an array is represented sometimes by putting parallel lines on either side of the array, or by writing det({array}).

Ok, you have defined these signed areas and so on to be determinants but what good is that?

All these quantities in all dimensions have some wonderful properties, which we can convert into properties of determinants, and we will be able to use them to calculate all of these. Not only that, we can calculate them, in any dimension, on a spreadsheet with only one non-trivial instruction, and some copying.

(On the Excel spreadsheet, there is a command, called mdeterm(), whose argument is an array, and which computes determinants, hence areas and volumes and so on. We can do the same without using this command.)


15.1 Represent the parallelepiped with corners at \((0, 0, 0)\) and adjacent corners at \((1, 2, 3), (1, 0, 1)\) and \((0, 1, 2)\) in two different ways.

15.2 Can you figure out the volume of this parallelepiped?

15.3 What is the relation of the area of a triangle to the area of a parallelogram having two of the sides of the triangle as sides of it?

15.4 If one corner of the parallelepiped described by the \(3\) by \(3\) array above is at the origin, what is the location of the "opposite" corner. (Figure out yourself what opposite means here.).

What are these "wonderful" properties?