Three perpendicular planes, and the eight regions they cut space into

What to assume they know

• Locating a point in a plane with two perpendicular axes, and reading a signed coordinate as a directed distance from an axis
• The four quadrants of the plane and the sign pattern that names each one
• That two distinct non-parallel planes meet in a line, and that a line can be perpendicular to a plane
• Right angles between lines in space, and the idea of two lines being perpendicular without meeting
• Counting by independent choices: two options taken three times over gives eight outcomes

What they should be able to do

• Explain why locating a fan tip or a hanging bulb inside a room takes three numbers rather than two, naming the room's floor and two adjacent walls as the three surfaces
• State what a rectangular coordinate system in space consists of, and say which of its parts are chosen and which are forced
• Derive each axis as the intersection of two of the three chosen planes
• Name all three coordinate planes and say which axis fails to lie in each
• State the sign convention this chapter adopts for each of the three directions, including which side of the YZ-plane is treated as the front
• Explain why three mutually perpendicular planes through one point cut space into exactly eight regions, using an independent-choice argument rather than a count off a diagram — and say why the perpendicularity cannot be dropped from the sentence
• Write the chapter's name for any octant, given the signs of a point inside it, and go back the other way
• Reproduce Table 11.1 from the two-line rule rather than from memory, and check it against the printed table
• Assign each of Exercise 11.1 q3's eight points to its octant and notice what the set of eight answers turns out to be
• Say what the Historical Note claims was missing between Descartes' work and Euler's

Where it usually goes wrong

• "Three planes always make eight pieces." They do not. Three planes sharing a single common line leave six wedges, and three parallel planes leave four slabs. The eight depends on the three being independent — which mutual perpendicularity guarantees. That is the actual work the perpendicularity does.
• "The axes come first and the planes are built from them." §11.2 goes the other way, and it matters, because the sign conventions are stated as sides of planes rather than as directions along lines. Both orders end in the same picture; only one of them makes the octant names read naturally.
• "An octant is a corner of a box." It is unbounded — it runs out forever in three directions. The box drawn in Fig 11.3 sits inside one octant and is not the octant.
• "Octants are numbered like binary digits." Not in this book. The plain binary reading would put (+, +, −) second; here it is fifth. The order is quadrant order first, then the z sign — which is why the two-line rule in section 9 works and a digit-counting rule does not.
• "Roman numerals and letter names are two different classifications." They are one classification written twice. Checking a few names against the table, rather than learning both lists, is the point of section 9.
• "The X-axis points to the right, as it does in a plane drawing." In Fig 11.1 the positive x direction is drawn toward the reader and the page is the XY-plane. A redraw that swaps this silently makes the printed sign conventions read backwards.
• "Space had to be described this way." The Historical Note exists to deny it. The idea sat unused for about three-quarters of a century after Descartes — 1637 to 1715 is 78 years — and the three coordinate planes in the form now taught arrive with Bernoulli's letter at the end of it.

Questions to check understanding

• Given a point's three signs, name its octant in Roman numerals and in the chapter's letter form
• Given an octant, write down a point inside it
• State how many regions three mutually perpendicular planes cut space into, and justify the number without drawing
• Name the coordinate plane spanned by a stated pair of axes, and the axis it omits
• Say which side of each coordinate plane the chapter counts as positive
• Identify the octant of a point after one of its coordinates is negated, and say which reflection that is
• Fill in blanks of the Exercise 11.1 q4 kind: which plane a named pair of axes determines, and how many octants there are

Examples worth working on the board

Inputs only. Anything marked verified is an added derivation from data printed inside pp. 208–216; the chapter's answers live in a separate file that was not opened.

• The two everyday locators (§11.1, p. 208). A ball in flight and an aeroplane in transit are offered as positions that move through space rather than across a sheet; then the lowest tip of a hanging bulb and the centre tip of a ceiling fan are offered as fixed positions inside a room. The room supplies the three surfaces: its floor and two walls that meet it. Use the room, not the aeroplane, to introduce the three planes — the room is already a rectangular coordinate system with the origin in a corner.
• Fig 11.1 (§11.2, p. 209), read off the printed page. Three sheets drawn through one point O: a horizontal parallelogram for the XY-plane, an upright rectangle carrying the Y and Z directions for the YZ-plane, and a narrow slanted parallelogram carrying the Z and X directions for the ZX-plane. Six labelled arrowheads: Z up, Z′ down, Y right, Y′ left, and — the detail a redraw usually gets wrong — X drawn to the lower left, coming toward the reader, with X′ going away to the upper right. Only O is lettered inside the figure; there is no point plotted in it.
• The sign conventions (§11.2, p. 209), one per plane, each phrased as a direction away from a plane: above the XY-plane is positive and below it negative; the side of the ZX-plane lying to the reader's right is positive, the other side negative; and the side of the YZ-plane nearer the reader counts as positive, the far side as negative. Verified consequence: each convention is a choice of one ray out of a pair, so there are 2³ = 8 ways the book could have set them, and Fig 11.1 shows the one it took.
• The eight octant names (§11.2, p. 209), paired here with the Roman numeral each carries: I is XOYZ, II is X′OYZ, III is X′OY′Z, IV is XOY′Z, V is XOYZ′, VI is X′OYZ′, VII is X′OY′Z′, and VIII is XOY′Z′. Verified: each name simply lists the three bounding rays, one from each axis, so the eight names are the eight ways of picking one ray per axis and none is repeated.
• Table 11.1 (Remark, §11.3, p. 210) — printed under §11.3 but it is §11.2's octants that it tabulates, so the explanation needs it here. Three rows against eight columns. Read off the printed page: the x row runs + − − + + − − +, the y row + + − − + + − −, and the z row + + + + − − − −. Verified structure: the z row is one block of four pluses and one of four minuses; within each block the (x, y) pairs run (+,+), (−,+), (−,−), (+,−) — which is the ordinary numbering of the four quadrants of a plane. So octants I–IV are quadrants I–IV raised above the XY-plane, and V–VIII are the same four dropped below it. Two sentences reconstruct all twenty-four cells.
• Example 2 (p. 211). Two points that differ in one sign only: (−3, 1, 2) and (−3, 1, −2). Verified: the first is octant II and the second octant VI. The pair is worth slowing down on, because it isolates exactly what the z sign changes and what it leaves alone.
• Exercise 11.1 q3 (p. 211), all eight items with their data intact: (1, 2, 3), (4, −2, 3), (4, −2, −5), (4, 2, −5), (−4, 2, −5), (−4, 2, 5), (−3, −1, 6), (−2, −4, −7). Verified: the octants are I, IV, VIII, V, VI, II, III and VII — that is, all eight octants, each hit exactly once, in a scrambled order. Say so at the end of the section; it turns a routine drill into a check that the classification is exhaustive.
• Exercise 11.1 q4(iii) (p. 211) asks for the number of parts the coordinate planes leave. Verified: eight, and section 7's argument is the reason rather than the table.
• The Historical Note (p. 216). Dates the explanation can use on a timeline: Descartes 1596–1650 with plane work in 1637; Fermat 1601–1665; La Hire 1640–1718; Bernoulli 1667–1748 writing to Leibnitz in 1715; Parent 1666–1716 presenting to the French Academy in 1700; Euler 1707–1783 publishing in 1748. Verified span: Descartes' 1637 plane geometry to Euler's 1748 treatment is 111 years, and Bernoulli's 1715 letter sits 78 years after 1637. The note's claim is that the three-dimensional idea was available to Descartes and simply not carried through.
• The portrait (p. 208). A framed engraving of Leonhard Euler with his dates, 1707–1783, set beside §11.1 — the same Euler the Historical Note closes on.

Figures to have open

• Fig 11.1 redrawn, with all six half-axes labelled and the positive x direction drawn toward the student as the chapter draws it. The chapter's own figure; sections 3 to 5 cannot be taught without it.
• An exploded build of the same figure: three sheets arriving one at a time, the crossing lines appearing as the second and third arrive. Standard schematic, and it is what makes section 2's claim visible.
• A binary answer tree of depth three whose eight leaves are then labelled with the eight octant names. Standard schematic; this is section 7's argument.
• Table 11.1 redrawn as a grid, with the z row's two blocks shaded differently and the repeated quadrant cycle marked underneath. Built from the chapter's own table.
• A four-date timeline for the Historical Note. Standard schematic; no portrait beyond the one the chapter already prints is needed.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 11 "Introduction to Three Dimensional Geometry", §11.1 Introduction, p. 208 — the room argument, the Euler portrait, and the footnote pointing at an NCERT laboratory-design handbook of 2005 (a pointer outside this textbook)
• §11.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space, p. 209 — Fig 11.1, the naming of the three planes, the sign conventions, the origin, the count of eight, and the eight octant names
• The Remark and Table 11.1, §11.3, p. 210 — the sign pattern of the eight octants
• Example 2 and Exercise 11.1 q3, q4(iii), p. 211
• Historical Note, p. 216
• Deliberate cross-reference outside this chapter: the four quadrants of the plane and the two-axis system §11.1 asks the reader to recall belong to the earlier coordinate work of this book, not to Chapter 11.