Why three num­bers?

Sup­pose a fly is buzzing around your class­room and you want to say exactly where it is. Two num­bers are not enough. You need three: how far it is from two of the walls, and how high it is above the floor. That is all three-dimen­sional geom­e­try really asks of you. We will set up axes in space, give names to the eight octants, stretch the dis­tance for­mula into three dimen­sions, and end with prac­tice on the chap­ter.

Axes, planes and octants

Draw three mutu­ally per­pen­dic­u­lar lines XOX′XOX', YOY′YOY', ZOZ′ZOZ' through the ori­gin OO; these are the coor­di­nate axes. Taken two at a time, they give three coor­di­nate planes: the XYXY-plane (z=0z = 0), the YZYZ-plane (x=0x = 0) and the ZXZX-plane (y=0y = 0), and these planes cut space into eight octants.

A point PP has coor­di­nates (x,y,z)(x, y, z), which are sim­ply its signed per­pen­dic­u­lar dis­tances from the YZYZ-, ZXZX- and XYXY-planes. So a point on the xx-axis looks like (x,0,0)(x, 0, 0), and a point on the XYXY-plane looks like (x,y,0)(x, y, 0).

Three perpendicular axes X, Y, Z through the origin O, with a point P(x, y, z) at the corner of a dashed box; its distances x, y and z from the three coordinate planes are highlighted.
A point in space is fixed by three coor­di­nates
Octant I II III IV V VI VII VIII
xx ++ −- −- ++ ++ −- −- ++
yy ++ ++ −- −- ++ ++ −- −-
zz ++ ++ ++ ++ −- −- −- −-

Exam­ple 1. (3,−2,5)(3, -2, 5) is in octant IV, (−1,−4,−2)(-1, -4, -2) is in VII, and (0,5,−3)(0, 5, -3) lies in the YZYZ-plane.

Dis­tance between two points

PQ=(x2−x1)2+(y2−y1)2+(z2−z1)2PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

This is noth­ing more than Pythago­ras used twice: once across the floor, and once going straight up. From the ori­gin it becomes OP=x2+y2+z2OP = \sqrt{x^2 + y^2 + z^2}.

Cuboid with P and Q at opposite corners and edges x2 - x1, y2 - y1 and z2 - z1; a dashed horizontal diagonal from P and the vertical edge z2 - z1 form a right triangle with PQ as hypotenuse.
PQ is the diag­o­nal of a box: Pythago­ras on the floor, then upwards

Exam­ple 2. P(1,−3,4)P(1, -3, 4), Q(−2,1,16)Q(-2, 1, 16): PQ=9+16+144=13PQ = \sqrt{9 + 16 + 144} = 13.

Exam­ple 3. A(1,5,2)A(1, 5, 2), B(2,4,6)B(2, 4, 6), C(5,7,6)C(5, 7, 6): AB=1+1+16=32AB = \sqrt{1 + 1 + 16} = 3\sqrt{2}, BC=9+9+0=32BC = \sqrt{9 + 9 + 0} = 3\sqrt{2}, AC=16+4+16=6AC = \sqrt{16 + 4 + 16} = 6. Two sides are equal, and since AB2+BC2=36=AC2AB^2 + BC^2 = 36 = AC^2, the tri­an­gle is right-angled isosce­les.

Exam­ple 4. A(1,2,3)A(1, 2, 3), B(3,6,7)B(3, 6, 7), C(4,8,9)C(4, 8, 9): AB=6AB = 6, BC=3BC = 3, and AC=9=AB+BCAC = 9 = AB + BC. The points must lie on one line, so they are collinear.

Exam­ple 5. Which points are equidis­tant from A(2,0,−1)A(2, 0, -1) and B(0,4,3)B(0, 4, 3)? Set­ting the dis­tances equal, (x−2)2+y2+(z+1)2=x2+(y−4)2+(z−3)2(x - 2)^2 + y^2 + (z + 1)^2 = x^2 + (y - 4)^2 + (z - 3)^2 gives x−2y−2z+5=0x - 2y - 2z + 5 = 0. That is a plane, the per­pen­dic­u­lar bisec­tor of ABAB.

Exam­ple 6. The set of points whose dis­tances from (3,0,0)(3, 0, 0) and (−3,0,0)(-3, 0, 0) add to 1010 is x225+y2+z216=1\displaystyle \tfrac{x^2}{25} + \tfrac{y^2 + z^2}{16} = 1.

Try these your­self

  1. Name the octant of each point: (−2,3,4)(-2, 3, 4); (4,−2,−5)(4, -2, -5); (−3,−1,6)(-3, -1, 6); (2,5,−1)(2, 5, -1).
  2. What are the coor­di­nates of a point on the yy-axis? In the ZXZX-plane?
  3. Find the dis­tance between (2,−1,3)(2, -1, 3) and (−4,1,0)(-4, 1, 0); between (a,b,c)(a, b, c) and (−a,−b,−c)(-a, -b, -c).
  4. Show that (−1,4,2)(-1, 4, 2), (1,3,0)(1, 3, 0) and (5,1,−4)(5, 1, -4) are collinear.
  5. Show that (1,2,3)(1, 2, 3), (3,2,5)(3, 2, 5) and (1,4,5)(1, 4, 5) form an equi­lat­eral tri­an­gle.
  6. Show that (0,0,0)(0, 0, 0), (2,1,1)(2, 1, 1), (3,3,2)(3, 3, 2) and (1,2,1)(1, 2, 1) form a par­al­lel­o­gram.
  7. Find the point on the xx-axis equidis­tant from (1,2,3)(1, 2, 3) and (3,5,−2)(3, 5, -2).
  8. Find the equa­tion of the set of points equidis­tant from (1,2,3)(1, 2, 3) and (3,2,−1)(3, 2, -1).
  9. Find the locus of points whose dis­tance from the ori­gin is 44.
  10. A box sits in the first octant with one cor­ner at the ori­gin and edges 33, 44 and 1212 along the axes. What is the length of the longest rod you can fit inside?

Answers to check against

Show answers
  1. II; VIII; III; V.
  2. (0,y,0)(0, y, 0); (x,0,z)(x, 0, z).
  3. 77; 2a2+b2+c22\sqrt{a^2 + b^2 + c^2}.
  4. The dis­tances are 33, 66, 99, and the two smaller ones add up to the largest.
  5. Each side is 222\sqrt{2}.
  6. Every side is 6\sqrt{6}, so oppo­site sides are equal; it is in fact a rhom­bus, and the diag­o­nals bisect each other at (1.5,1.5,1)(1.5, 1.5, 1).
  7. (x−1)2+13=(x−3)2+29(x - 1)^2 + 13 = (x - 3)^2 + 29 gives 4x=244x = 24, x=6x = 6: (6,0,0)(6, 0, 0).
  8. x−2z=0x - 2z = 0.
  9. It is the sphere x2+y2+z2=16x^2 + y^2 + z^2 = 16.
  10. 9+16+144=13\sqrt{9 + 16 + 144} = 13.