Answer:
We interpret a determinant as the area of a parallelogram, and a determinant as the volume of a parallelepiped.
DET-0070: Determinants as Areas and Volumes
Determinants can be interpreted geometrically as areas and volumes. This might make intuitive sense if we observe that is the area of a parallelogram determined by and .
We are used to working with column vectors. In this section, however, when we use vectors to form a matrix we will regard them as row vectors. So, the vector forms the first row, , of the matrix .
If we double one of the vectors, the determinant doubles, and so does the area of the parallelogram.
If we shear the figure, the area is not affected (why?), but neither is the determinant.
Three-by-three determinants can be interpreted as volumes in a similar way. We will now proceed to derive the relationship between determinants, area and volume more formally.
Determinant and the Area of a Parallelogram
Consider the parallelogram determined by vectors and in .
Recall that the area of a parallelogram is given by the product of the length of the base and the height. As shown in the diagram below, the length of the base is the magnitude of . The height, , can be found using trigonometry
Using the area of a parallelogram formula together with Theorem th:crossproductsin we get We have established the following formula.
Find the area of the parallelogram in the diagram.
The vectors that determine the parallelogram are
The problem we run into is that these vectors are in , whereas the cross product is
defined only for vectors in . We will get around this difficulty by “padding” our
vectors with zeros on the bottom. In other words, we will consider them as vectors
sitting in the -coordinate plane in . This allows us to compute the cross
product
The area of the parallelogram is then given by
Example ex:areaofparallelogram illustrates an important phenomenon. Observe that the zeros in the last column of the determinant ensure that the and components of the cross product are zero, while the last component is the determinant of the matrix whose rows are the two vectors that determine the parallelogram in . In general, if the parallelogram is determined by vectors then the area of the parallelogram can be computed as follows:
So the area of the parallelogram turns out to be the absolute value of the determinant of the matrix whose rows are the two vectors that determine the parallelogram. The following formula summarizes our discussion.
Determinant and the Volume of a Parallelepiped
Our next goal is to find the volume of a three-dimensional figure called a parallelepiped. A parallelepiped is a six-faced figure whose opposite faces are congruent parallelograms located in parallel planes. A parallelepiped is a three-dimensional counterpart of a parallelogram, and is determined by three non-coplanar vectors in . The figure below shows a parallelepiped determined by three vectors.
Consider a parallelepiped determined by vectors , and , as shown below.
The volume of a parallelepiped is given by We will consider the parallelogram determined by and to be the base of the parallelepiped. Thus, the area of the base is given by
The height of the parallelepiped is measured along a line perpendicular to the base. By Theorem th:crossproductorthtouandv, lies on such a line. Let be the angle between and , . Then the height, , of the parallelepiped is given by This gives us the following formula for the volume of the parallelepiped
We have established the following formula.
Our next goal is to show that this expression for the volume is equal to the determinant of a matrix whose rows are the vectors that determine the parallelepiped.
Let then
The expression in (eq:boxproduct) is sometimes referred to as the box product.
Practice Problems
Let be a square determined by and . Let be a parallelogram determined by vectors
and . Sketch both figures in the same coordinate plane, and use geometry to
explain why and have the same area. Compute the area of using Formula
form:areaofparallelogramdeterminant.