660 CHAPTER 9 Polar Coordinates; Vectors Rectangular Coordinates in Space In a plane, each point is associated with an ordered pair of real numbers. In space, each point is associated with an ordered triple of real numbers.Through a fixed point, called the origin O, draw three mutually perpendicular lines: the x-axis, the y-axis, and the z-axis. On each of these axes, select an appropriate scale and the positive direction. See Figure 78. The direction chosen for the positive z-axis in Figure 78 makes the system righthanded. This conforms to the right-hand rule, which states that if the index finger of the right hand points in the direction of the positive x-axis and the middle finger points in the direction of the positive y-axis, then the thumb will point in the direction of the positive z-axis. See Figure 79. Figure 78 22 2 4 2 22 22 4 2 4 O y x z Figure 79 O y x z Figure 80 2 4 2 (2, 0, 0) (0, 0, 4) (2, 3, 4) (0, 3, 0) (2, 3, 0) 4 6 8 2 4 y x z NOTE The equation y b = represents a horizontal line in 2-space and it represents a plane in 3-space. j Figure 81 3 Plane z 5 3 Plane y 5 4 4 (b) y x z x 5 0 yz-plane z 5 0 xy-plane y 5 0 xz-plane (a) x z y Associate with each point P an ordered triple x y z , , ( ) of real numbers, the coordinates of P. For example, the point 2, 3, 4 ( ) is located by starting at the origin and moving 2 units along the positive x-axis, 3 units in the direction of the positive y-axis, and 4 units in the direction of the positive z-axis. See Figure 80. Figure 80 also shows the location of the points 2, 0, 0 , 0, 3, 0 , ( ) ( ) 0, 0, 4 , ( ) and 2, 3, 0 ( ). Points of the form x, 0, 0 ( ) lie on the x-axis, and points of the forms y 0, , 0 ( ) and z 0, 0, ( ) lie on the y-axis and z-axis, respectively. Points of the form x y , , 0 ( ) lie in a plane called the xy-plane. Its equation is z 0. = Similarly, points of the form x z , 0, ( ) lie in the xz-plane (equation y 0 = ), and points of the form y z 0, , ( ) lie in the yz-plane (equation x 0 = ). See Figure 81(a). By extension of these ideas, all points obeying the equation z 3 = will lie in a plane parallel to and 3 units above the xy-plane. The equation y 4 = represents a plane parallel to the xz-plane and 4 units to the right of the plane y 0. = See Figure 81(b). Now Work PROBLEM 9 1 Find the Distance between Two Points in Space The formula for the distance between two points in space is an extension of the Distance Formula for points in the plane given in Section 1.2.
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