17.1 Signed Areas and Volumes

Area, like distance, and volume in customary language are quantities that are always positive. However, we will find it useful to give signs to them.

Thus if you are driving a car, and another car is 2 car lengths ahead of you, you might assign a positive distance to the distance between your car and it, and if it is behind you we can assign a negative distance to the same.

The same sort of thing can be done with area and volume. If you have an x-axis, you can assign positive area to area above it, and negative area to that below it. And the same idea can be generalized to volumes, in three dimensions, and even further.

Why would you want to do this?

If you plot the distance between you and an oncoming vehicle, when you are standing still, this distance will decrease as it approaches, and then increase again, after it goes past you. Thus the plot of its distance will look like a V. If we use signed distance, and the vehicle is moving at a uniform speed, the distance from where you were before you dived out of the way will be a straight line. After it passes you its distance becomes negative. Straight lines are so much easier to deal with than V-like curves that we prefer to deal with them.

The area of a rectangle, as I hope you remember, is the product of the lengths of its sides, if we ignore signs, which we normally do. This is the basic fact we start from.
Similarly, the volume of a cube is the cube of its side length. The analogue of a rectangle in three dimensions is called a "rectangular parallelepiped" and its volume is the product of the lengths of its three sides. And you can imagine similar statements in more dimensions.

We will now discuss the areas of tilted parallelograms, and the volumes of general parallelepipeds, which are three dimensional six sided figures whose opposite sides are parallel to one another.

Why?

You will soon see why. Be patient and you will learn something you do not now know.