Three Dimensional Space

This section only expects students to have familiarity with measuring distance in two dimensions. This can likely be covered in around 40 minutes and should offer opportunities for users to get used to the 3D interactable plots and the structure of in class activities.

Some instructors may want to emphasize the idea of right and left handed coordinates more than others. In particular, many students in computer science have used left handed coordinate systems as is used in many game engines. Similarly, many physics and engineering students will be expected to use an orientation similar to .

While the discussion of how coordinates are measured in two dimensional rectangular coordinates is an idea that many students think they can skip past, we are using this idea to make sure everyone is on common ground with the particular geometric measurements that define coordinates. Many students struggle with later work on coordinates and coordinate systems because they do not understand the underlying geometric foundation.

The discussion of octants is another place where some instructors may skip over this idea, but we have elected to use this as a way to generalize the very useful idea of quadrants from the Cartesian plane. Similarly, activities like ask student to attempt to draw in three dimensions. While this is usually very difficult, some instructors may use these kinds of activities as a step toward growing students ability to draw and communicate about figures in 3D, while other instructors may omit these tasks entirely. I personally (this is Nick talking) have found tremendous benefit to spending time and effort in getting students (and me) to work on drawing and understanding figures in three dimensions. Talking about surface and directional derivatives is significantly easier if everyone has some similar mental pictures and can represent those key ideas in their own hand. I think this argument applies to the descriptions of fundamental planes given in this section. The TL;DR version is that while you can save time by not doing most of this section, the cost is that many students will struggle with more difficult visualization and communication tasks.

The functions you are most familiar with have the form \(y=f(x)\), where \(x\) is the input or domain variable and \(y\) is viewed as the output variable. The graphs of these functions are plotted on a coordinate plane with the input expressed in terms of the horizontal coordinate and the output as the vertical coordinate. The following preview activity asks you to recall some algebraic and geometric ideas from before your single variable calculus course.

As you read in the introduction to this chapter, we will need to expand beyond the idea of measuring change involving expressions with one number as an input or output. One of the new kinds of functions we will examine is of the form \(z=g(x,y)\), where the input is an ordered pair, \((x,y)\), and the output of the function \(g\) is the real number \(z\). In order to graph such a function, we will need to represent the basic ideas of coordinates and functions in a more general setting to accommodate more dimensions.

The tasks in connect ideas used in coordinates and graphing for two dimensions to specific measurements. We will use these connections to build useful tools for three or more dimensions. In the rest of this section, we will define the measurements for coordinates and basic measurements in three dimensions.

We let \(\R^2\) denote the set of all ordered pairs of real numbers in the plane (two copies of the real number system) and let \(\R^3\) represent the set of all ordered triples of real numbers, which constitutes three-dimensional space. For example, \((1,\pi)\) is in \(\R^2\) and \((-e^2,\frac{\sqrt{7}}{4},0)\) is in \(\R^3\), but \((\ln(3),2i)\) is not in \(\R^2\) because the second coordinate, \(2i\), is an imaginary number and not a real number.

To plot points in three-dimensional space, we will set up a coordinate system with three mutually perpendicular axesthe \(x\)-axis, the \(y\)-axis, and the \(z\)-axis (called the coordinate axes). There are essentially two different ways we could set up a 3D coordinate system, as shown in Figure, which depicts only the positive portions of the axes. Thus, before we can proceed, we need to establish a convention.

The distinction between these two figures is subtle, but important. In the coordinate system shown in , imagine that you are sitting on the positive \(z\)-axis next to the label \(z\). Looking down at the \(x\)- and \(y\)-axes, you see that the \(y\)-axis is obtained by rotating the \(x\)-axis by \(90^\circ\) in the clockwise direction. Again sitting on the positive \(z\)-axis in the coordinate system at right in , you see that the \(y\)-axis is obtained by rotating the \(x\)-axis by \(90^\circ\) in the counterclockwise direction.

We call the coordinate system in a right-hand system; if we point the index finger of our right hand along the positive \(x\)-axis and our middle finger along the positive \(y\)-axis, then our thumb points in the direction of the positive \(z\)-axis. Following mathematical conventions, we choose to use a right-hand system throughout this book.

We let \(\R^2\) denote the set of all ordered pairs of real numbers in the plane (two copies of the real number system) and let \(\R^3\) represent the set of all ordered triples of real numbers (which constitutes three-space). For example, \((1,\pi)\) is in \(\R^2\) and \((-e^2,\frac{\sqrt{7}}{4},0)\) is in \(\R^3\), but \((\ln(3),2i)\) is not in \(\R^2\) because the second coordinate is not a real number because \(2i\) is imaginary.

In other disciplines or settings you may see the \(y\)-axis as being oriented vertically with the \(x\)-axis going left/right and the \(z\)-axis going in and out of the page like in the figure below. This orientation may be useful for you because you can think of the traditional orientation of the \(xy\)-plane (when drawn on paper) with the positive \(z\)-axis coming out of the page.

Now that we have established how our axes will be defined, we can use these axes to define how to measure coordinates and make a few basic graphs.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

At this point, we are ready to understand graphs of some simple equations in three dimensions. For example, in \(\R^2\), the graphs of the equations \(x=a\) and \(y=b\), where \(a\) and \(b\) are constants, are lines parallel to the coordinate axes. Remember that a graph of an equation is a plot of all points that satisfy this equation. This means that the graph of an equation is a visual representation of all locations whose coordinates will make the left side of your equation equal to the right side of the equation. The equation should give you a way to test whether a point is on the graph or not.

For instance, the graph of \(x=2\) in two dimensions will be a vertical line with \(x\)-intercept of 2, because the points of the form \((2,y)\) satisfy the equation \(x=2\) (for any choice of \(y \in \mathbb{R}\)) as shown in . In the next activity we consider the three-dimensional analogs of these kinds equations.

illustrates that equations where one of the rectangular coordinates is held constant lead to planes parallel to the coordinate planes. When we make the constant 0, we get the coordinate planes themselves . The \(xy\)-plane satisfies \(z=0\), the \(xz\)-plane satisfies \(y=0\), and the \(yz\)-plane satisfies \(x=0\) (see ). Planes of the form \(x=a\), \(y=b\), \(z=c\) are called fundamental planes are useful in understanding and building structures in three dimensions. You can see how the intersection of fundamental planes \(x=1\), \(y=-2\), and \(z=3\) corresponds to the point \((1,-2,3)\) in . As always, you should rotate and change the viewpoint of the figure to make sure you understand how each fundamental plane and the point \(P\) are oriented relative to the axes.

In a plot like , you see a grid that helps measure coordinates of points in 2D. In you can look at how a different number of fundamental planes forming a grid looks. You can use the slider to increase or decrease the number of fundamental planes displayed in the grid. When you have more than one plane in each direction there will be a lot of visual clutter and very hard to distinguish fine features, especially something in the middle of the plot. Contrast this to how helpful the grid in is.

We conclude this section by using our knowledge of how to measure straight-line distance in \(\R^2\) to find a formula for distance in three dimensions. On a related note, we define a circle in \(\R^2\) as the set of all points equidistant from a fixed point. In \(\R^3\), we call the set of all points equidistant from a fixed point a sphere. To find the equation of a sphere, we need to understand how to calculate the distance between two points in three-space, and we explore this idea in the next activity.

The method used in does not depend on anything but the coordinates between the two points, so we can use the last result to measure the distance between any two points in \(\mathbb{R}^3\).

The distance between points \(P=(x_0, y_0, z_0)\) and \(Q=(x_1, y_1, z_1)\) (denoted as \(|PQ|\)) in \(\R^3\) is given by the formula \[\begin{aligned}\end{aligned}\]

As showed, the distance in two or three (or more!) dimensions depends on the change in each coordinate from one point to the other. Note that the distance does not depend on whether we consider \(P\) to \(Q\) or \(Q\) to \(P\).

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.