Section 12.1: Three dimenstional Coordinate Geometry

To locate a point in a plane, we need two numbers. We know that any point in the plane can be represented as an ordered pair \((a, b)\) of real numbers, where \(a\) is the x-coordinate and \(b\) is the y-coordinate. For this reason, a plane is called two-dimensional. To locate a point in space, three numbers are required. We represent any point in space by an ordered triple \((a, b, c)\) of real numbers.

In order to represent points in space, we first choose a fixed point \(O\) (the origin) and three directed lines through \(O\) that are perpendicular to each other, called the coordinate axes and labeled the x-axis, y-axis, and z-axis. Usually we think of the x- and y-axes as being horizontal and the z-axis as being vertical, and we draw the orientation of the axes as in Figure 1.

The direction of the z-axis is determined by the right-hand rule as illustrated in Figure 2: If you curl the fingers of your right hand around the z-axis in the direction of a 90° counterclockwise rotation from the positive x-axis to the positive y-axis, then your thumb points in the positive direction of the z-axis.

The three coordinate axes determine the three coordinate planes illustrated in Figure 3(a). The xy-plane is the plane that contains the x- and y-axes; the yz-plane contains the y- and z-axes; the xz-plane contains the x- and z-axes. These three coordinate planes divide space into eight parts, called octants. The first octant, in the foreground, is determined by the positive axes.

Because many people have some difficulty visualizing diagrams of three-dimensional figures, you may find it helpful to do the following [see Figure 3(b)]. Look at any bottom corner of a room and call the corner the origin. The wall on your left is in the xz-plane, the wall on your right is in the yz-plane, and the floor is in the xy-plane. The x-axis runs along the intersection of the floor and the left wall. The y-axis runs along the intersection of the floor and the right wall. The z-axis runs up from the floor toward the ceiling along the intersection of the two walls. You are situated in the first octant, and you can now imagine seven other rooms situated in the other seven octants (three on the same floor and four on the floor below), all connected by the common corner point \(O\).

Now if \(P\) is any point in space, let \(a\) be the (directed) distance from the yz-plane to \(P\), let \(b\) be the distance from the xz-plane to \(P\), and let \(c\) be the distance from the xy-plane to \(P\). We represent the point \(P\) by the ordered triple \((a, b, c)\) of real numbers and we call \(a, b,\) and \(c\) the coordinates of \(P\); \(a\) is the x-coordinate, \(b\) is the y-coordinate, and \(c\) is the z-coordinate. Thus, to locate the point \((a, b, c)\), we can start at the origin \(O\) and move \(a\) units along the x-axis, then \(b\) units parallel to the y-axis, and then \(c\) units parallel to the z-axis as in Figure 4.

The point \(P(a, b, c)\) determines a rectangular box as in Figure 5. If we drop a perpendicular from \(P\) to the xy-plane, we get a point \(Q\) with coordinates \((a, b, 0)\) called the projection of \(P\) onto the xy-plane. Similarly, \(R(0, b, c)\) and \(S(a, 0, c)\) are the projections of \(P\) onto the yz-plane and xz-plane, respectively.

As numerical illustrations, the points \((-4, 3, -5)\) and \((3, -2, -6)\) are plotted in Figure 6.

The Cartesian product \(\mathbb{R} \times \mathbb{R} \times \mathbb{R} = \{(x, y, z) | x, y, z \in \mathbb{R}\}\) is the set of all ordered triples of real numbers and is denoted by \(\mathbb{R}^3\). We have given a one-to-one correspondence between points \(P\) in space and ordered triples \((a, b, c)\) in \(\mathbb{R}^3\). It is called a three-dimensional rectangular coordinate system. Notice that, in terms of coordinates, the first octant can be described as the set of points whose coordinates are all positive.

In two-dimensional analytic geometry, the graph of an equation involving \(x\) and \(y\) is a curve in \(\mathbb{R}^2\). In three-dimensional analytic geometry, an equation in \(x, y,\) and \(z\) represents a surface in \(\mathbb{R}^3\).

EXAMPLE 1 What surfaces in \(\mathbb{R}^3\) are represented by the following equations? (a) \(z = 3\) (b) \(y = 5\)

In general, if \(k\) is a constant, then \(x = k\) represents a plane parallel to the yz-plane, \(y = k\) is a plane parallel to the xz-plane, and \(z = k\) is a plane parallel to the xy-plane. In Figure 5, the faces of the rectangular box are formed by the three coordinate planes \(x = 0\) (the yz-plane), \(y = 0\) (the xz-plane), and \(z = 0\) (the xy-plane), and the planes \(x = a, y = b,\) and \(z = c\).

EXAMPLE 2 (a) Which points \((x, y, z)\) satisfy the equations \(x^2 + y^2 = 1\) and \(z = 3\)? (b) What does the equation \(x^2 + y^2 = 1\) represent as a surface in \(\mathbb{R}^3\)?

EXAMPLE 3 Describe and sketch the surface in \(\mathbb{R}^3\) represented by the equation \(y = x\).

The familiar formula for the distance between two points in a plane is easily extended to the following three-dimensional formula.

Distance Formula in Three Dimensions The distance \(|P_1P_2|\) between the points \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\) is \[ |P_1P_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

To see why this formula is true, we construct a rectangular box as in Figure 11, where \(P_1\) and \(P_2\) are opposite vertices and the faces of the box are parallel to the coordinate planes. If \(A(x_2, y_1, z_1)\) and \(B(x_2, y_2, z_1)\) are the vertices of the box indicated in the figure, then \[ |P_1A| = |x_2 - x_1| \quad |AB| = |y_2 - y_1| \quad |BP_2| = |z_2 - z_1| \] Because triangles \(P_1BP_2\) and \(P_1AB\) are both right-angled, two applications of the Pythagorean Theorem give \[ |P_1P_2|^2 = |P_1B|^2 + |BP_2|^2 \] and \[ |P_1B|^2 = |P_1A|^2 + |AB|^2 \] Combining these equations, we get \[ |P_1P_2|^2 = |P_1A|^2 + |AB|^2 + |BP_2|^2 = |x_2 - x_1|^2 + |y_2 - y_1|^2 + |z_2 - z_1|^2 \] \[ = (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2 \] Therefore \[ |P_1P_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

EXAMPLE 4 The distance from the point \(P(2, -1, 7)\) to the point \(Q(1, -3, 5)\) is \[ |PQ| = \sqrt{(1 - 2)^2 + (-3 - (-1))^2 + (5 - 7)^2} = \sqrt{1 + 4 + 4} = 3 \]

EXAMPLE 5 Find an equation of a sphere with radius \(r\) and center \(C(h, k, l)\).

Equation of a Sphere An equation of a sphere with center \(C(h, k, l)\) and radius \(r\) is \[ (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \] In particular, if the center is the origin \(O\), then an equation of the sphere is \[ x^2 + y^2 + z^2 = r^2 \]

EXAMPLE 6 Show that \(x^2 + y^2 + z^2 + 4x - 6y + 2z + 6 = 0\) is the equation of a sphere, and find its center and radius.

EXAMPLE 7 What region in \(\mathbb{R}^3\) is represented by the following inequalities? \[ 1 \le x^2 + y^2 + z^2 \le 4 \quad z \le 0 \]

What does the equation \(x = 4\) represent in \(\mathbb{R}^2\)? What does it represent in \(\mathbb{R}^3\)? Illustrate with sketches.

What does the equation \(y = 3\) represent in \(\mathbb{R}^3\)? What does \(z = 5\) represent? What does the pair of equations \(y = 3, z = 5\) represent? In other words, describe the set of points \((x, y, z)\) such that \(y = 3\) and \(z = 5\). Illustrate with a sketch.

Describe and sketch the surface in \(\mathbb{R}^3\) represented by the equation \(x + y = 2\).

Describe and sketch the surface in \(\mathbb{R}^3\) represented by the equation \(x^2 + z^2 = 9\).

Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? \(P(3, -2, -3), Q(7, 0, 1), R(1, 2, 1)\)

Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? \(P(2, -1, 0), Q(4, 1, 1), R(4, -5, 4)\)

Determine whether the points lie on a straight line. (a) \(A(2, 4, 2), B(3, 7, -2), C(1, 3, 3)\) (b) \(D(0, -5, 5), E(1, -2, 4), F(3, 4, 2)\)

Show that the equation \(x^2 + y^2 + z^2 - 2x - 4y + 8z = 15\) represents a sphere, and find its center and radius.

Show that the equation \(x^2 + y^2 + z^2 + 8x - 6y + 2z + 17 = 0\) represents a sphere, and find its center and radius.

Show that the equation \(2x^2 + 2y^2 + 2z^2 = 8x - 24z + 1\) represents a sphere, and find its center and radius.

Show that the equation \(3x^2 + 3y^2 + 3z^2 = 10 + 6y + 12z\) represents a sphere, and find its center and radius.

1. Prove that the midpoint of the line segment from \(P_1(x_1, y_1, z_1)\) to \(P_2(x_2, y_2, z_2)\) is \[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2} \right) \]
2. Find the lengths of the medians of the triangle with vertices A(1, 2, 3), B(-2, 0, 5), and C(4, 1, 5). (A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side.)

Describe in words the region of \(\mathbb{R}^3\) represented by the equation \(x = 5\).

Describe in words the region of \(\mathbb{R}^3\) represented by the equation \(y = -2\).

Describe in words the region of \(\mathbb{R}^3\) represented by the inequality \(y < 8\).

Describe in words the region of \(\mathbb{R}^3\) represented by the inequality \(z \ge -1\).

Describe in words the region of \(\mathbb{R}^3\) represented by the inequality \(0 \le z \le 6\).

Describe in words the region of \(\mathbb{R}^3\) represented by the equation \(y^2 = 4\).

Describe in words the region of \(\mathbb{R}^3\) represented by the equations \(x^2 + y^2 = 4, z = -1\).

Describe in words the region of \(\mathbb{R}^3\) represented by the equation \(x^2 + y^2 = 4\).

Describe in words the region of \(\mathbb{R}^3\) represented by the equation \(x^2 + y^2 + z^2 = 4\).

Describe in words the region of \(\mathbb{R}^3\) represented by the inequality \(x^2 + y^2 + z^2 \le 4\).

Describe in words the region of \(\mathbb{R}^3\) represented by the inequality \(1 \le x^2 + y^2 + z^2 \le 5\).

Describe in words the region of \(\mathbb{R}^3\) represented by the equation \(x = z\).

Describe in words the region of \(\mathbb{R}^3\) represented by the inequality \(x^2 + z^2 \le 9\).

Describe in words the region of \(\mathbb{R}^3\) represented by the inequality \(x^2 + y^2 + z^2 > 2z\).

Write inequalities to describe the region: The region between the yz-plane and the vertical plane \(x = 5\).

Write inequalities to describe the region: The solid cylinder that lies on or below the plane \(z = 8\) and on or above the disk in the xy-plane with center the origin and radius 2.

Write inequalities to describe the region: The region consisting of all points between (but not on) the spheres of radius \(r\) and \(R\) centered at the origin, where \(r < R\).

The figure shows a line \(L_1\) in space and a second line \(L_2\), which is the projection of \(L_1\) onto the xy-plane. (In other words, the points on \(L_2\) are directly beneath, or above, the points on \(L_1\).) (a) Find the coordinates of the point P on the line \(L_1\). (b) Locate on the diagram the points A, B, and C, where the line \(L_1\) intersects the xy-plane, the yz-plane, and the xz-plane, respectively.

Find the volume of the solid that lies inside both of the spheres \(x^2 + y^2 + z^2 + 4x - 2y + 4z + 5 = 0\) and \(x^2 + y^2 + z^2 = 4\)

Find the distance between the spheres \(x^2 + y^2 + z^2 = 4\) and \(x^2 + y^2 + z^2 = 4x + 4y + 4z - 11\).