this is for college students who wants to know more about integrals

1. 1.1 The 3-Dimensional Coordinate System
(MATH 71-Analytic Geometry and Calculus 3)
NESTOR G. ACALA, PhD
Mathematics Department
Mindanao State University Main Campus
Marawi City
nestor.acala@gmail.com

2. CURVES AND SURFACES IN R3
The Three-Dimensional Coordinate System
The three coordinate axes ( x-axis, y-axis, z-axis)

3. CURVES AND SURFACES IN R3
The Three-Dimensional Coordinate System
The three coordinate axes ( x-axis, y-axis, z-axis)determine the three coordinate
planes:

4. CURVES AND SURFACES IN R3
The Three-Dimensional Coordinate System
The three coordinate axes ( x-axis, y-axis, z-axis)determine the three coordinate
planes:xy−plane which contains the x and y axes,

5. CURVES AND SURFACES IN R3
The Three-Dimensional Coordinate System
The three coordinate axes ( x-axis, y-axis, z-axis)determine the three coordinate
planes:xy−plane which contains the x and y axes,yz− plane which contains the
y and z axes

6. CURVES AND SURFACES IN R3
The Three-Dimensional Coordinate System
The three coordinate axes ( x-axis, y-axis, z-axis)determine the three coordinate
planes:xy−plane which contains the x and y axes,yz− plane which contains the
y and z axes and the xz− plane which contains the x and z axes.

7. CURVES AND SURFACES IN R3
The Three-Dimensional Coordinate System
The three coordinate axes ( x-axis, y-axis, z-axis)determine the three coordinate
planes:xy−plane which contains the x and y axes,yz− plane which contains the
y and z axes and the xz− plane which contains the x and z axes.The three
coordinate planes divide the space into eight parts called octants. The rst
octant, in the foreground is determined by the positive axes.

8. x

9. x
y

10. x
y
z

11. x
y
z
•

12. x
y
z
•
origin or pole

13. R3
= R × R × R = {(x, y, z) : x, y, z ∈ R}.
Points on the
• x−axis: (a, 0, 0)
• y−axis: (0, b, 0)
• z−axis: (0, 0, c)
• xy−plane: (a, b, 0)
• xz−plane: (a, 0, c)
• yz−plane: (0, b, c).

14. Plotting points on the xy-plane

15. Plotting points on the xy-plane
x

16. Plotting points on the xy-plane
x
y

17. Plotting points on the xy-plane
x
y
z

18. Plotting points on the xy-plane
x
y

19. Plotting points on the xy-plane
x
y

20. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)

21. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)

22. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)

23. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)

24. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)

25. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)

26. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)

27. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)

28. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)

29. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)
•
(−3, 2, 0)

30. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)
•
(−3, 2, 0)
•
(3, 0, 0)

31. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)
•
(−3, 2, 0)
•
(3, 0, 0)
•
(0, 3, 0)

32. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)
•
(−3, 2, 0)
•
(3, 0, 0)
•
(0, 3, 0)

33. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)
•
(−3, 2, 0)
•
(3, 0, 0)
•
(0, 3, 0)

34. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)
•
(−3, 2, 0)
•
(3, 0, 0)
•
(0, 3, 0)
•
(3, 3, 0)

35. Plotting points on the xy-plane
x
y
Plot the points (1, −2, 0), (−3, 2, 0) and (3, 3, 0)
•
(1, 0, 0)
•
(0, −2, 0)
•
(1, −2, 0)
•
(−3, 0, 0)
•
(0, 2, 0)
•
(−3, 2, 0)
•
(3, 0, 0)
•
(0, 3, 0)
•
(3, 3, 0)
z

36. Plotting points on the xz-plane

37. Plotting points on the xz-plane
x

38. Plotting points on the xz-plane
x
z

39. Plotting points on the xz-plane
x
z

40. Plotting points on the xz-plane
x
z

41. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)

42. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)

43. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)

44. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)

45. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)

46. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)
•
(3, 0, 2)

47. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)
•
(3, 0, 2)
•
(−1, 0, 0)

48. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)
•
(3, 0, 2)
•
(−1, 0, 0)
•
(0, 0, −2)

49. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)
•
(3, 0, 2)
•
(−1, 0, 0)
•
(0, 0, −2)

50. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)
•
(3, 0, 2)
•
(−1, 0, 0)
•
(0, 0, −2)

51. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)
•
(3, 0, 2)
•
(−1, 0, 0)
•
(0, 0, −2)
• (−1, 0, −2)

52. Plotting points on the xz-plane
x
z
Plot the points (3, 0, 2) and (−1, 0, −2)
•
(3, 0, 0)
•
(0, 0, 2)
•
(3, 0, 2)
•
(−1, 0, 0)
•
(0, 0, −2)
• (−1, 0, −2)
y

53. Plotting points on the yz-plane
y

54. Plotting points on the yz-plane
y
z

55. Plotting points on the yz-plane
y
z

56. Plotting points on the yz-plane
y
z

57. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)

58. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)

59. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)

60. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)

61. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)

62. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)

63. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)

64. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)

65. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)

66. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)

67. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)
•
(0, −2, 1)
•
(0, −1, 0)

68. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)
•
(0, −2, 1)
•
(0, −1, 0)
•
(0, 0, 2)

69. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)
•
(0, −2, 1)
•
(0, −1, 0)
•
(0, 0, 2)

70. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)
•
(0, −2, 1)
•
(0, −1, 0)
•
(0, 0, 2)

71. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)
•
(0, −2, 1)
•
(0, −1, 0)
•
(0, 0, 2)
• (0, 1, 2)

72. Plotting points on the yz-plane
y
z
Plot the points (0, 2, −1), (0, −2, 1) and (0, 1, 2)
•
(0, 2, 0)
•
(0, 0, −1)
• (0, 2, −1)
•
(0, −2, 0)
•
(0, 0, 1)
•
(0, −2, 1)
•
(0, −1, 0)
•
(0, 0, 2)
• (0, 1, 2)
x

73. Projection of a Point on the Coordinate Planes
A point P(a, b, c) on R3 determines a rectangular box (see gure in the next
slide). Dropping a perpendicular from P to the xy−plane, we get a point
Q(a, b, 0) called the projection of P on the xy− plane. Similarly, R(0, b, c) and
S(a, 0, c) are the projections of P on the yz− and xz− planes respectively.

74. x
y
z
•
P(a, b, c)

75. x
y
z
•
P(a, b, c)

76. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)

77. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)

78. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)

79. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)

80. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)

81. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)

82. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)

83. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)

84. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)

85. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)

86. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)

87. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)
• Q(0, b, c)

88. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)
• Q(0, b, c)

89. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)
• Q(0, b, c)

90. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)
• Q(0, b, c)

91. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)
• Q(0, b, c)
•
R(a, 0, c)

92. x
y
z
•
P(a, b, c)
•
A(a, 0, 0)
• B(0, b, 0)
• Q(a, b, 0)
• C(0, 0, c)
• Q(0, b, c)
•
R(a, 0, c)

93. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z

94. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)

95. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)

96. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)

97. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)

98. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)

99. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)

100. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)

101. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)

102. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units

103. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)

104. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)

105. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)

106. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)
•
R(0, −2, 3)

107. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)
•
R(0, −2, 3)

108. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)
•
R(0, −2, 3)

109. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)
•
R(0, −2, 3)

110. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)
•
R(0, −2, 3)
• S(2, 0, 3)

111. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)
•
R(0, −2, 3)
• S(2, 0, 3)

112. Example: Plot the point P(2, −2, 3) on 3D coordinate system.
x
y
z
•
A(2, 0, 0)
•
B(0, −2, 0)
•
Q(2, −2, 0)
•
P(2, −2, 3)
3 units
• C(0, 0, 3)
•
R(0, −2, 3)
• S(2, 0, 3)

113. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z

114. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)

115. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)

116. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)

117. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)

118. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)

119. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)
•
Q(−2, 2, 0)

120. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)
•
Q(−2, 2, 0)

121. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)
•
Q(−2, 2, 0)
•
P(−2, 2, −3)

122. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)
•
Q(−2, 2, 0)
•
P(−2, 2, −3)
3 units

123. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)
•
Q(−2, 2, 0)
•
P(−2, 2, −3)
3 units

124. Example: Plot the point P(−2, 2, −3) on 3D coordinate system.
x
y
z
•
A(−2, 0, 0)
•
B(0, 2, 0)
•
Q(−2, 2, 0)
•
P(−2, 2, −3)
3 units
•
C(0, 0, −3)

125. Remark
In the two-dimensional analytic geometry, the graph of a nondegenerate equation
involving x and y is a curve in R2. In the three-dimensional analytic geometry,
nondegenerate equation in x, y and z represents a surface in R3.

126. Remark
In the two-dimensional analytic geometry, the graph of a nondegenerate equation
involving x and y is a curve in R2. In the three-dimensional analytic geometry,
nondegenerate equation in x, y and z represents a surface in R3.
Example: What surfaces are represented by the following equation in R3?
1. y = 2 2. z = 3

127. x
y
z
y = 2
xz−plane
•
(0, 2, 0)

128. x
y
z
xy−plane
•
(0, 0, 3)
z = 3

129. Remark
In general, equation of the form ax + by + cz + d = 0, where a, b, and c are not
all zero represents a plane in R3. In particular, if k ∈ R, x = k, y = k, z = k
represent planes parallel to yz−, xz− and xy− planes respectively.

130. Distance Formulas in R3
Distance Formula
The distance |P1P2| between the points P1(x1.y1, z1) and P2(x2, y2, z2) is given
by
d = |P1P2| =
p
(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2.

131. Distance Formulas in R3
Distance Formula
The distance |P1P2| between the points P1(x1.y1, z1) and P2(x2, y2, z2) is given
by
d = |P1P2| =
p
(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2.
Midpoint Formula in R3
The midpoint (x, y, z) between the points P1(x1.y1, z1) and P2(x2, y2, z2) is
given by
(x, y, z) =
x1 + x2
2
,
y1 + y2
2
,
z1 + z2
2
.

132. Example: Find the distance between the points P(1, −2, 3) and
Q(2, 0, −1).
Solution:

133. Example: Find the distance between the points P(1, −2, 3) and
Q(2, 0, −1).
Solution: Choosing (x1, y1, z1) = (1, −2, 3) and (x2, y2, z2) = (2, 0, −1),

134. Example: Find the distance between the points P(1, −2, 3) and
Q(2, 0, −1).
Solution: Choosing (x1, y1, z1) = (1, −2, 3) and (x2, y2, z2) = (2, 0, −1),we
obtain
d =
p
(2 − 1)2 + (0 − (−2))2 + (−1 − 3)2
=
p
12 + 22 + (−4)2
=
√
21 units.

135. Example: Find the distance between the points P(1, −2, 3) and
Q(2, 0, −1).
Solution: Choosing (x1, y1, z1) = (1, −2, 3) and (x2, y2, z2) = (2, 0, −1),we
obtain
d =
p
(2 − 1)2 + (0 − (−2))2 + (−1 − 3)2
=
p
12 + 22 + (−4)2
=
√
21 units.

136. Distance of a Point from a Plane
The distance of a point P(x0, y0, z0) from the plane φ : Ax + By + Cz + D = 0
is given by
d =
|Ax0 + By0 + Cz0 + D|
√
A2 + B2 + C2
.

137. Example: Find the distance of the point P(−2, 1, −1) from the
plane 4x − 3y = 4.
Solution:

138. Example: Find the distance of the point P(−2, 1, −1) from the
plane 4x − 3y = 4.
Solution:In this case, the equation of the plane in standard form is

139. Example: Find the distance of the point P(−2, 1, −1) from the
plane 4x − 3y = 4.
Solution:In this case, the equation of the plane in standard form is
φ : 4x − 3y + 0z − 4 = 0 (A = 4, B = −3, C = 0, D = −4).
Thus, the distance of the point (−2, 1, −1) from the plane φ is

140. Example: Find the distance of the point P(−2, 1, −1) from the
plane 4x − 3y = 4.
Solution:In this case, the equation of the plane in standard form is
φ : 4x − 3y + 0z − 4 = 0 (A = 4, B = −3, C = 0, D = −4).
Thus, the distance of the point (−2, 1, −1) from the plane φ is
d =
|4(−2) + (−3)(1) + 0(−1) + (−4)|
p
42 + (−3)2 + 02
=
| − 8 − 3 − 4|
√
16 + 9
=
| − 15|
√
25
=
15
5
= 3 units.

141. Example: Find the distance of the point P(−2, 1, −1) from the
plane 4x − 3y = 4.
Solution:In this case, the equation of the plane in standard form is
φ : 4x − 3y + 0z − 4 = 0 (A = 4, B = −3, C = 0, D = −4).
Thus, the distance of the point (−2, 1, −1) from the plane φ is
d =
|4(−2) + (−3)(1) + 0(−1) + (−4)|
p
42 + (−3)2 + 02
=
| − 8 − 3 − 4|
√
16 + 9
=
| − 15|
√
25
=
15
5
= 3 units.