Definition 2.1.3. Distance In Space.
Let \(P=(x_1,y_1,z_1)\) and \(Q = (x_2,y_2,z_2)\) be points in space. The distance \(D\) between \(P\) and \(Q\) is
\begin{equation*}
D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\text{.}
\end{equation*}
Section 2.1 Introduction to Cartesian Coordinates in Space
We reviewed the two-dimensional Cartesian Plane in Section 1.3. This is the familiar background on which much of your high school mathematics played out, and it provides the setting for the Calculus of one variable encountered in an introductory calculus course.
While there is wonderful mathematics to explore in “2D,” we live in a “3D” world and eventually we will want to apply mathematics involving this third dimension. In this section we introduce Cartesian coordinates in space and explore basic surfaces. This will lay a foundation for much of what we do in the remainder of the text.
Each point \(P\) in space can be represented with an ordered triple, \(P=(a,b,c)\text{,}\) where \(a\text{,}\) \(b\) and \(c\) represent the relative position of \(P\) along the \(x\)-, \(y\)- and \(z\)-axes, respectively. Each axis is perpendicular to the other two.
Visualizing points in space on paper can be problematic, as we are trying to represent a 3-dimensional concept on a 2-dimensional medium. We cannot draw three lines representing the three axes in which each line is perpendicular to the other two. Despite this issue, standard conventions exist for plotting shapes in space that we will discuss that are more than adequate.
One convention is that the axes must conform to the right hand rule. This rule states that when the index finger of the right hand is extended in the direction of the positive \(x\) axis, and the middle finger (bent “inward” so it is perpendicular to the palm) points along the positive \(y\) axis, then the extended thumb will point in the direction of the positive \(z\) axis. (It may take some thought to verify this, but this system is inherently different from the one created by using the “left hand rule.”)
As long as the coordinate axes are positioned so that they follow this rule, it does not matter how the axes are drawn on paper. There are two popular methods that we briefly discuss.
In Figure 2.1.1 we see the point \(P=(2,1,3)\) plotted on a set of axes. The basic convention here is that the \(xy\)-plane is drawn in its standard way, with the \(z\)-axis down to the left. The perspective is that the paper represents the \(xy\)-plane and the positive \(z\) axis is coming up, off the page. This method is preferred by many engineers. Because it can be hard to tell where a single point lies in relation to all the axes, dashed lines have been added to let one see how far along each axis the point lies.
One can also consider the \(xy\)-plane as being a horizontal plane in, say, a room, where the positive \(z\)-axis is pointing up. When one steps back and looks at this room, one might draw the axes as shown in Figure 2.1.2. The same point \(P\) is drawn, again with dashed lines. This point of view is preferred by most mathematicians, and is the convention adopted by this text.
Just as the \(x\)- and \(y\)-axes divide the plane into four quadrants, the \(x\)-, \(y\)-, and \(z\)-coordinate planes divide space into eight octants. The octant in which \(x\text{,}\) \(y\text{,}\) and \(z\) are positive is called the first octant. We do not name the other seven octants in this text.
Subsection 2.1.1 Measuring Distances
It is of critical importance to know how to measure distances between points in space. The formula for doing so is based on measuring distance in the plane, and is known (in both contexts) as the Euclidean measure of distance.
We refer to the line segment that connects points \(P\) and \(Q\) in space as \(\overline{PQ}\text{,}\) and refer to the length of this segment as \(\norm{\overline{PQ}}\text{.}\) The above distance formula allows us to compute the length of this segment.
Example 2.1.4. Length of a line segment.
Let \(P = (1,4,-1)\) and let \(Q = (2,1,1)\text{.}\) Draw the line segment \(\overline{PQ}\) and find its length.
Solution.
The points \(P\) and \(Q\) are plotted in Figure 2.1.5; no special consideration need be made to draw the line segment connecting these two points; simply connect them with a straight line. One cannot actually measure this line on the page and deduce anything meaningful; its true length must be measured analytically. Applying Definition 2.1.3, we have
\begin{equation*}
\norm{\overline{PQ}} = \sqrt{(2-1)^2+(1-4)^2+(1-(-1))^2} = \sqrt{14}\approx 3.74\text{.}
\end{equation*}
Subsection 2.1.2 Introduction to Planes in Space
The coordinate axes naturally define three planes (shown in Figure 2.1.6), the coordinate planes: the \(xy\)-plane, the \(yz\)-plane and the \(xz\)-plane. The \(xy\)-plane is characterized as the set of all points in space where the \(z\)-value is 0. This, in fact, gives us an equation that describes this plane: \(z=0\text{.}\) Likewise, the \(xz\)-plane is all points where the \(y\)-value is 0, characterized by \(y=0\text{.}\)
The equation \(x=2\) describes all points in space where the \(x\)-value is 2. This is a plane, parallel to the \(yz\)-coordinate plane, shown in Figure 2.1.7.
Example 2.1.8. Regions defined by planes.
Sketch the region defined by the inequalities \(-1\leq y\leq 2\text{.}\)
Solution.
The region is all points between the planes \(y=-1\) and \(y=2\text{.}\) These planes are sketched in Figure 2.1.9, which are parallel to the \(xz\)-plane. Thus the region extends infinitely in the \(x\) and \(z\) directions, and is bounded by planes in the \(y\) direction.
Exercises 2.1.3 Exercises
1.
The points \(A=(1,4,2)\text{,}\) \(B=(2,6,3)\) and \(C=(4,3,1)\) form a triangle in space. Find the distances between each pair of points and determine if the triangle is a right triangle.
\(\left\lVert\overline{AB}\right\rVert=\)
\(\left\lVert\overline{BC}\right\rVert=\)
\(\left\lVert\overline{CA}\right\rVert=\)
The three points
- do
- do not
form a right triangle.
2.
The points \(A=(1,1,3)\text{,}\) \(B=(3,2,7)\text{,}\) \(C=(2,0,8)\) and \(D = (0,-1,4)\) form a quadrilateral \(ABCD\) in space. Is this a parallelogram?
Exercise Group.
In the following exercises, describe the region in space defined by the inequalities.
3.
\(0\leq x\leq 3\)
4.
\(x\geq 0,\ y\geq0, \ z\geq0\)
5.
\(y\geq 3\)
