10.1 Rectangular Coordinates in Space

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A two-dimensional rectangular coordinate system is represented by a plane in which every point $P$ corresponds to exactly one ordered pair of real numbers $( x,\,y) ,$ the coordinates of $P.$ In space every point $P$ corresponds to exactly one ordered triple of real numbers $( x,y,z)$.

1 Locate Points in Space

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We begin by selecting a fixed point called the origin $O$. Through the origin, draw three mutually perpendicular number lines, called the coordinate axes. The coordinate axes are usually labeled the ${x}$-axis the ${y}$-axis and the ${z}$-axis. On each axis choose one direction as positive and select an appropriate scale, as shown in Figure 1.

Notice label in Figure 1 that the positive $z$-axis points upward, the positive $y$-axis points toward the right, and the positive $x$-axis points forward. This is known as a right-handed system because it conforms to the right-hand rule. The right-hand rule states: if the index finger of the right hand points in the direction of the positive $x$-axis and the other fingers point in the direction of the positive $y$-axis, then the thumb will point in the direction of the positive $z$-axis, as shown in Figure 2.

As in the plane, we assign coordinates to each point $P$ in space, but in space we use an ordered triple of real numbers $( x,\,y,\,z)$. For example, the point $( 3,5,7)$, is the point for which $3$ is the $x$-coordinate, $5$ is the $y$-coordinate, and $7$ is the $z$-coordinate. The point $( 3,5,7)$ is plotted by starting at the origin, moving 3 units along the positive $x$-axis, $5$ units in the direction of the positive $y$-axis, and $7$ units in the direction of the positive $z$-axis. See Figure 3.

Figure 3 also shows the points $( 3,0,0)$, $(0,5,0)$, $(0,0,7)$, $(3,5,0)$, $( 3,0,7)$, $0,5,7)$ and $(0,0,0),$ which form a box with the points $(0,0,0)$ and $( 3,5,7)$ as opposite vertices.

Points of the form $( x,0,0)$ lie on the $x$-axis, while points of the form $( 0,y,0)$ and $( 0,0,z)$ lie on the $y$-axis and $z$-axis, respectively.

Points of the form $(x,y,0)$ lie on a plane called the $xy$-plane. The $xy$-plane is perpendicular to the $z$-axis and its equation is $z =0$. This is the plane used in the familiar two-dimensional rectangular coordinate system.

Points of the form $( x,0,z)$ lie on the $xz$-plane. The $xz$-plane is perpendicular to the $y$-axis, and its equation is $y=0$. Finally, points of the form $( 0,y,z)$ lie on the $yz$-plane, which is perpendicular to the $x$-axis, and its equation is $x=0$. Collectively, these three planes are referred to as coordinate planes. See Figure 4.

In general, for any real number $k$, the graph of $x=k$ is a plane parallel to the $yz$-plane; the graph of $y=k$ is a plane parallel to the $xz$-plane; and the graph of $z=k$ is a plane parallel to the $xy$-plane. As shown in Figure 5, the graph of the equation $z=5$ is a plane parallel to and $5$ units above the $xy$-plane.

2 Find the Distance Between Two Points in Space

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The formula for the distance between two points in space is an extension of the distance formula in the plane.

To find the distance $|P_{1}~P_{2}|$ between two points $P_{1}=(x_{1},y_{1},z_{1})$ and $P_{2}=(x_{2},y_{2},z_{2})$, we use the Pythagorean Theorem twice. Figure 6 shows the points $P_{1}$ and $P_{2}$ and a third point $A=(x_{2},y_{2},z_{1})$. The first application of the Pythagorean Theorem involves observing that the triangle $P_{1}~A~P_{2}$ is a right triangle where the side of length $|P_{1}~P_{2}|$ is the hypotenuse. As a result, $$\left\vert P_{1}~P_{2}\right\vert =\sqrt{\vert P_{1}~A\vert ^{2}+\vert A~P_{2}\vert ^{2}}\qquad{\color{#0066A7}{\hbox{Pythagorean Theorem}}}$$

The points $P_{1}$ and $A$ lie in a plane parallel to the $xy$-plane. Do you see why? To find $\left\vert P_{1}~A\right\vert$, we use the distance formula in the plane: $$\begin{equation*} \left\vert P_{1}\,A\right\vert =\sqrt{( x_{2}-x_{1}) ^{2}+( y_{2}-y_{1}) ^{2}} \end{equation*}$$

The points $P_{2}$ and $A$ lie on a line parallel to the $z$-axis, so that $\left\vert A~P_{2}\right\vert =\left\vert z_{2}-z_{1}\right\vert$ and $\left\vert A~P_{2}\right\vert ^{2}=(z_{2}-z_{1})^{2}.$ Combining these results, we obtain a formula for the distance between two points in space.

3 Find the Equation of a Sphere

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The collection of all points in space that are a fixed distance $R$ from a fixed point $P_{0}=( x_{0},y_{0},z_{0})$ is called a sphere. See Figure 7. The fixed distance $R$ is called the radius, and the fixed point $P_{0}$ is called the center of the sphere. The distance from any point $P=(x, y, z)$ on a sphere of radius $R$ to the center point $P_{0}=(x_{0},y_{0},z_{0})$ is $\vert PP_{0}\vert=R$. The Distance Formula in space shows that $$\begin{equation*} \vert P\!P_{0}\vert =\sqrt{ (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}=R \end{equation*}$$

Squaring both sides gives the standard form of the equation of a sphere.