Math 152 { c Lynch 1 of 6 Section 11.1 { Three-Dimensional Coordinate Systems Any point in a plane can be represented by an ordered pair (a; b) of real numbers, where a is the x-coordinate and b is the y-coordinate. A point in three dimensional space is represented by an ordered triple (a; b; c) of real numbers. Definitions. 1. To locate a point in three-dimensional 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. Then each point in the three-dimensional space can be represented by an ordered triple (a; b; c) where a, b, and c are the x-, y- and z-coordinates respectively. 2. The right-hand rule determines the direction of the z-axis. If you curl the fingers of your right hand around the z-axis in the direction from the positve x-axis to the positive y-axis, then your thumb points in the positive direction of the z-axis. 3. The three coordinate axes determine the three coordinate planes shown below. The xy-plane is the plane that contains the x- and y-axes. Similarly, the xz-plane is the plane that contains the x- and z-axes, and the yz-plane is the plane that contains the y- and z-axes. The coordinate planes divide space into eight parts, called octants. The first octant is the octant where all the coordinate axes are positive. Math 152 { c Lynch Section 11.1 2 of 6 4. Let P be any point in space with a the directed distance from P to the yz-plane, b the directed distance from P to the xz-plane, and c the directed distance from P to the xy-plane. Then we represent P by the ordered triple (a; b; c) and a, b, and c are the coordinates of P ; a is the x-coordinate, b is the y-coordinate, and c is the z-coordinate. The first octant is the point where all the coordinates are positive. 5. The point P determines a rectangular box as shown above. If we drop a per- pendicular line from the point P (a; b; c) to the xy-plane, we get a point Q with coordinates (a; b; 0) called the projection of P on the xy-plane. Similarly, the points R(0; b; c) and S(a; 0; c) are the projections of P on the yz-plane and xz- plane, respectively. 6. The Cartesian product R × R × R = f(x; y; z) j x; y; z 2 Rg is the set of all ordered triples of real numbers and is denoted R3. We have given a one-to-one correspondence between points in space and ordered triples in R3. This is called a three-dimensional rectangular coordinate system. (Note that R represents the real numbers which correspond to a one-dimensional line, and R2 represents ordered pairs of real numbers which correspond to the normal two-dimension plane. ) Example 1. Plot the points (−4; 3; 5) and (3; −2; −6). Example 2. What is the projection of the point (−4; 3; 5) onto the xy-plane, the xz-plane, and the yz-plane? Math 152 { c Lynch Section 11.1 3 of 6 In two-dimensional geometry, the graph of an equation involving x and y is a curve in R2. In three-dimensional geometry, an equation in x, y, and z represents a surface in R3. Example 3. What surfaces in R3 are represented by the following equations? (a) z = 5 (b) x = −3 Note. The context of an equation will determine whether the equation represents a curve in R2 or a surface in R3. For comparison graph x = −3 in R2. Example 4. Describe and sketch the surface in R3 represented by the equation y = x. Math 152 { c Lynch Section 11.1 4 of 6 Distance Formula in Three Dimensions. The distance jP1P2j between the points P1(x1; y1; z1) and P2(x2; y2; z2) is p 2 2 2 jP1P2j = (x1 − x2) + (y1 − y2) + (z1 − z2) Example 5. Find the lengths of the sides of the triangle ABC with A(3; −4; 1), B(5; −3; 0), and C(6; −7; 4). Determine if the triangle is isosceles, a right triangle, both, or neither. Example 6. Find an equation of the sphere with radius r and center C(h; k; l). Math 152 { c Lynch Section 11.1 5 of 6 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 = r2: In particular, if the center is the origin O, then an equation of a sphere is x2 + y2 + z2 = r2: Example 7. Find the equation of a sphere if one of its diameters has endpoints (3; −4; 1) and (5; −3; 0). (Note we calculated this distance in Example 5.) Example 8. What is the intersection of the sphere from Example 7 with the xy-plane? Example 9. Find the equation of a sphere with center (−1; 4; 0) if it touches the xz-plane. Example 10. Write the equation of the sphere given by x2 −4x+y2 +2y +z2 −3z = 4 in standard form. Find the center and radius. Math 152 { c Lynch Section 11.1 6 of 6 Example 11. Describe the regions of R3 represented by the following inequalities. (a) z < 1 (b) x2 + y2 = 4 (c) x2 + 2x + y2 + z2 − 4z ≥ 4.