Reading a triple as three perpendicular distances, one per plane

What to assume they know

• The rectangular coordinate system of §11.2: three mutually perpendicular planes, the three axes they cross in, the origin, and the sign convention on each direction (Three perpendicular planes, and the eight regions they cut space into)
• Dropping a perpendicular from a point to a plane, and that the foot is unique
• Dropping a perpendicular from a point to a line in a plane
• That a line perpendicular to a plane is perpendicular to every line drawn in that plane through its foot
• Reading a signed coordinate in the plane as a directed distance
• What it means for two sets to be in one-to-one correspondence

What they should be able to do

• State the two claims §11.3 has to prove, and say how one construction run in both directions supplies both
• Carry out the Fig 11.2 construction on a stated point: perpendicular to the XY-plane, then perpendicular from that foot to the x-axis
• Identify OL, LM and MP in Fig 11.2 with the three coordinates, and write down the triple of the intermediate foot M
• Run the construction backwards: from a given triple, locate the point on the x-axis, then the point in the XY-plane, then the point in space
• Describe Fig 11.3's alternative construction and name the three planes it draws
• List the eight vertices of the Fig 11.3 box with their triples, and identify the three faces the chapter names by their vertex letters
• State which coordinate plane each of the three coordinates is measured from, and explain why the pairing is not the one a student first guesses
• Explain why a negative coordinate is still a statement about a distance
• Say what §11.3's one-to-one correspondence actually asserts, pairing the points of space with ordered triples of real numbers, and which half of it each direction of the construction establishes

Where it usually goes wrong

• **"x is the distance from the x-axis."** It is not, and this is the single most common error in the chapter. The distance from the x-axis mixes the other two coordinates together; x is the distance from the YZ-plane, the plane that the x-axis pierces at the origin. The test question to ask: which of the three planes does not contain the x-axis?
• "The three numbers are just labels, so the order is a convention." The order is a convention only in the sense that the three planes were named in some order. Once named, each slot has a fixed meaning as a distance from a specific plane, and swapping two entries moves the point.
• "Fig 11.2 and Fig 11.3 are two drawings of the same argument." They are two different routes to the same triple. Fig 11.2's construction, run forwards and then backwards, is what earns the correspondence — a pairing that goes one way only is not one-to-one. Fig 11.3 adds nothing to that and instead supplies the reading §11.3 closes on, in which each coordinate is a distance from a plane.
• "P is the corner of the box, so the box is the point." The box is scaffolding. Nothing about the box is part of the answer except the three edge lengths meeting at O; slide P and the box changes shape while the meaning of the triple does not.
• "A perpendicular distance cannot be negative, so coordinates must be positive." The chapter's own figures answer this: the drawn point is deliberately placed in the all-positive octant, and the text says at once that the other octants flip the signs. A coordinate reports the size of the distance and the side of the plane at the same time.
• "M is just a construction mark." M is a genuine point of the XY-plane with its own triple, printed in the figure. Recognising M as (x, y, 0) is the first instance of the zero-coordinate reading the next topic runs on.
• "Three planes through P are needed to define the point." They are needed to recover it from three numbers. Fig 11.2 shows two perpendiculars are enough in the other direction. Which construction you need depends on which way you are going.

Questions to check understanding

• Given a point's triple, state its perpendicular distance from each of the three coordinate planes
• Given a point, write the triple of its foot on the XY-plane
• Name which coordinate plane a stated coordinate is measured from, and which axis that plane fails to contain
• Reconstruct the Fig 11.2 construction in words from a labelled diagram
• On a lettered box like Fig 11.3, given P's triple, write the triples of the other seven corners
• Say what would go wrong if only one of §11.3's two constructions were available
• Explain in words why a coordinate can be negative although it is called a distance

Examples worth working on the board

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

• Fig 11.2 (§11.3, p. 209), read off the printed page and a close-up. Axes Z upward, Y to the right, X toward the lower left. P is lettered at the top right with its triple (x, y, z) beside it. A vertical segment runs down from P to M, and M carries its own printed label (x, y, 0). L sits on the x-axis with the segment OL marked x; the segment LM is marked y; MP is marked z.
• The letter C in Fig 11.2, and why no triple can be given for it. Measured on p. 209: C is lettered on the y-axis, exactly where the drawn vertical from M up to P crosses that axis. Nothing else meets C. There is no segment from C to M, and nothing anywhere in this figure is drawn parallel to the x-axis, so O, L, M and C are not drawn as a parallelogram.

The measurement settles what C is not. LM is drawn horizontally, parallel to the drawn y-axis, so if C were the point (0, y, 0) then OC would have to come out equal to LM by construction. Measured, OC is 298 px against LM's 461 px. **C is therefore not (0, y, 0)**, and in three dimensions the segment from M to P is parallel to the z-axis and meets the y-axis only when the first coordinate is zero. What the page letters is a crossing in the projection, not a point the construction produces. No triple can be assigned to it, and an explanation must not supply one.

• The Fig 11.2 construction, stated as three quantities. OL is the first coordinate, LM the second, MP the third. Verified: M is the foot of the perpendicular from P to the XY-plane, so M and P agree in their first two coordinates and M's third is zero — which is exactly the label the figure prints on M. Handing the explanation M's triple as well as P's is what lets section 4 make the point that the construction produces a second, simpler point on the way.
• The reverse construction (§11.3, p. 210). From a triple: fix the point on the x-axis at the first entry; move off it parallel to the y-axis to reach the point of the XY-plane with the first two entries; then raise a perpendicular to the XY-plane and go the third entry along it. Verified consequence: every step is forced once the previous one is done, which is why the point the triple names is unique and not merely available.
• Fig 11.3 (§11.3, p. 210), read off the printed page and a close-up. A box drawn with one corner at O and the opposite corner at P, with three planes through P parallel to the coordinate planes cutting the axes at A, B and C. The dashed measurements inside the figure are marked x along the x-axis, y along the y-axis and z up the z-axis. The other lettered corners are D, E and F. The chapter names three of the box's faces by their four corners: ADPF, BDPE and CEPF. Verified vertex list, from the construction: A = (x, 0, 0), B = (0, y, 0), C = (0, 0, z), D = (x, y, 0), E = (0, y, z), F = (x, 0, z), P = (x, y, z), and O the origin. Verified face check: ADPF is the set of corners whose first entry is x, BDPE those whose second is y, and CEPF those whose third is z — so each named face is one of the three planes drawn through P, and the three of them meet only at P.
• The closing identification (§11.3, p. 210). The three coordinates are the perpendicular distances from the YZ-plane, the ZX-plane and the XY-plane, in that order. Verified against Fig 11.3: the distance from P to the YZ-plane is the length OA, which the figure marks x; from P to the ZX-plane is OB, marked y; from P to the XY-plane is OC, marked z. The Summary on p. 215 repeats the same pairing, so it is not a passing remark.
• Why the axis reading and the plane reading agree. Verified: in Fig 11.2 the first coordinate arrives as OL, a length measured along the x-axis from the origin; in Fig 11.3 it arrives as the distance from P to the YZ-plane. These are the same number because the YZ-plane cuts the x-axis at the origin at right angles. Saying this out loud is what stops the two figures looking like two unrelated definitions.
• The all-positive case is a special case. §11.3 states that the point drawn in Fig 11.2 sits in the octant with all three signs positive, and that other octants change the signs. Verified: the figure therefore proves the construction only for one eighth of space; the other seven follow by the same construction with the measurement taken along the opposite ray, which is what the sign conventions of §11.2 are for.
• Example 1 (p. 211) uses Fig 11.3 with P at (2, 4, 5) and asks for F. It is worked in Which coordinate goes to zero on which axis or plane as a zero-coordinate question; mention it here only to show that the lettered box is used again and is worth learning properly.

Figures to have open

• Fig 11.2 redrawn with every lettering the printed figure carries, C on the y-axis included, and with M's printed triple kept. **C must be drawn where the page puts it — on the crossing of the M-to-P vertical with the y-axis — and must not be given a triple.** The chapter's own figure; sections 2, 4 and 12 depend on the lettering being complete and on that restraint.
• Fig 11.3 redrawn as a box on O and P with all eight corners lettered A to F plus O and P, and the three named faces shadeable one at a time. The chapter's own figure and the spine of sections 6 to 8.
• A movement of the reverse construction: a point sliding along the x-axis, then out parallel to the y-axis, then up. Standard schematic; it carries section 5.
• A single panel putting the axis reading and the plane reading of the same coordinate side by side, with the right angle at the origin marked. Standard schematic; it is the answer to the chapter's most-missed point.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 11 "Introduction to Three Dimensional Geometry", §11.3 Coordinates of a Point in Space, pp. 209–210 — Fig 11.2, the two-perpendicular construction, the reverse construction, Fig 11.3, the three named faces, and the closing identification of the coordinates with three perpendicular distances
• §11.2, p. 209 — the coordinate planes, the origin and the sign conventions this section takes as settled
• Example 1, p. 211 — the first use of the Fig 11.3 lettering
• Summary, p. 215 — restates the same pairing of each coordinate with the plane it is measured from
• Deliberate cross-reference outside this chapter: the perpendicularity facts used in section 2 — that the foot of a perpendicular to a plane is unique, and that such a perpendicular meets every line of the plane through the foot at right angles — belong to school solid geometry and are assumed rather than proved here.