Which coordinate goes to zero on which axis or plane

What to assume they know

• That each coordinate is the perpendicular distance from a specific coordinate plane, and which plane goes with which coordinate (Reading a triple as three perpendicular distances, one per plane)
• The three coordinate planes and the three axes of §11.2, and the origin as their common point (Three perpendicular planes, and the eight regions they cut space into)
• The lettered box of Fig 11.3 and the triples of its corners
• That two coordinate planes meet in an axis, and that all three meet only at the origin
• Reading a plane, a line and a point as objects of decreasing freedom

What they should be able to do

• State what a single zero coordinate asserts about the point, in terms of a plane
• Say why a point on an axis must have two zero coordinates, and name them
• Write the general form of a point on each of the three axes and in each of the three coordinate planes
• Give the triple of the origin and explain why nothing shorter than three zeros pins it
• Predict, from the number of zeros in a triple, whether the point lies in an octant, on a coordinate plane, on an axis, or at the origin
• Work Example 1: read off the coordinates of a named corner of the Fig 11.3 box by deciding which measurement vanishes there
• List the triples of all eight corners of that box and sort them by how many zeros they carry
• Answer Exercise 11.1 q1 and q2 and explain why they are the same question in different clothing
• Complete Exercise 11.1 q4's three blanks by reasoning rather than recall
• Notice that the chapter writes the same plane as ZX in the text and XZ in an exercise, and say why the answer is unaffected

Where it usually goes wrong

• "A zero means that coordinate is missing or unknown." It is the most definite value a coordinate can take: it pins the point onto a named plane. Everything else leaves the point free to slide.
• **"A point on the x-axis has x equal to zero."** Exactly backwards, and it is the error Exercise 11.1 q1 is built to catch. On the x-axis the first coordinate is the only one that survives; the other two die.
• "One zero puts the point on an axis." One zero puts it on a plane, which is a much weaker statement — the point still has two free coordinates. Two zeros are what an axis costs.
• "The origin is just where the drawing starts." It is the only point of space with three zeros, and it is the only point lying on all three coordinate planes at once. Both descriptions pick out the same single point, and section 4 should show they are the same statement.
• "XZ-plane and ZX-plane are different planes." They are the same plane written with the letters in either order. What is not interchangeable is which slot of the triple vanishes on it, and that is settled by the coordinate-to-plane pairing, not by the spelling.
• "Points with a zero are edge cases you can ignore." Every axis, every coordinate plane and the origin itself are made entirely of such points, and every one of the box corners in Fig 11.3 except P is one. They are most of the furniture, not the exceptions.
• "The Summary lists everything worth keeping." It gives the three axis forms and no plane form. A student revising from p. 215 alone will not have the fact that Exercise 11.1 q4(ii) asks for.

Questions to check understanding

• Given that a point lies on a named axis, state which of its coordinates are zero
• Given that a point lies in a named coordinate plane, state which coordinate is zero and what can be said about the others
• Write the general form of a point on each axis and in each coordinate plane
• Given a triple containing zeros, say whether the point is in an octant, on a plane, on an axis, or at the origin
• Read the coordinates of a named corner of a lettered box drawn on a general point
• Fill in blanks of the Exercise 11.1 q4 kind
• Explain why a point of the x-axis cannot have a non-zero third coordinate

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.

• The Note (§11.3, p. 210). Three statements in one shaded box: the origin's triple is all zeros; a point of the x-axis has its second and third entries zero; a point of the YZ-plane has its first entry zero. Read it off the printed page — it is a boxed insert and the box is easy to lose in extraction. Verified pattern: the box gives one example of each of the three zero counts, so it is already a complete demonstration of section 5's claim; the chapter simply does not say so.
• The Summary's axis forms (p. 215). Three lines, one per axis: a point of the x-axis, a point of the y-axis, a point of the z-axis, each written with two of the three slots set to zero. Verified: in each case the two zeros are the coordinates measured from the two planes that cross in that axis, and the surviving entry is the one measured from the plane the axis pierces. The Summary gives the three axis forms and gives no plane form — checked against p. 215 — although the Note on p. 210 supplies one for the YZ-plane. Section 11 is where the explanation restores the missing two.
• Example 1 (p. 211). Working in Fig 11.3 with P at (2, 4, 5), the question asks for the coordinates of the corner F. The printed reasoning is that the measurement along the y-direction is zero there, so F is (2, 0, 5). Verified: F is the corner of the box lying in the ZX-plane, and the middle slot is precisely the distance from the ZX-plane, so it must vanish.
• All eight corners of the Fig 11.3 box, with P at (2, 4, 5). Verified from the construction of §11.3 and checked against the chapter's three printed face names ADPF, BDPE and CEPF: O is (0, 0, 0); A is (2, 0, 0) on the x-axis; B is (0, 4, 0) on the y-axis; C is (0, 0, 5) on the z-axis; D is (2, 4, 0) in the XY-plane; E is (0, 4, 5) in the YZ-plane; F is (2, 0, 5) in the ZX-plane; and P is (2, 4, 5) with no zero at all. Verified count: the eight corners carry three zeros once, two zeros three times, one zero three times and no zero once. One box exhibits every zero pattern exactly once, which is why it is the right figure for this topic even though it was drawn for the previous one.
• Exercise 11.1 q1 (p. 211) places a point somewhere along the x-axis and asks for its remaining two coordinates. Verified: both zero, because the x-axis lies in the XY-plane and in the ZX-plane at once, and those are the two planes the second and third coordinates are measured from.
• Exercise 11.1 q2 (p. 211) places a point somewhere in the XZ-plane and asks what follows for its middle coordinate. Verified: it is zero, and nothing at all can be said about the other two. Pair this with q1: q1 gives a line and gets two zeros, q2 gives a plane and gets one.
• Exercise 11.1 q4 (p. 211), three blanks. (i) the plane the x- and y-axes determine between them; (ii) the shape of a triple belonging to a point of the XY-plane; (iii) how many octants the coordinate planes leave. Verified: the XY-plane; a triple whose third entry is zero; and eight. Item (ii) is the plane form the Summary omits, so the exercise is quietly supplying what the Summary drops.
• The naming inconsistency. Verified by reading the pages: the running text of §11.2 and the Summary on p. 215 write this plane as the ZX-plane, while Exercise 11.1 q2 on p. 211 writes it as the XZ-plane. Both name the plane spanned by the same two axes, and the coordinate that vanishes on it is the same one either way. Worth thirty seconds, because a student who has learnt the coordinate order by position rather than by plane will hesitate.

Figures to have open

• The Fig 11.3 box with all eight corners lettered and each carrying its triple for P = (2, 4, 5), so that the zeros can be highlighted in place. The chapter's own figure with coordinates added; sections 6 and 7 run on it.
• A three-plane diagram in which planes can be lit individually, so a point can be shown lying on one, then two, then all three. Standard schematic; it carries sections 2 to 4.
• A 1–3–3–1 layout of the eight corners grouped by zero count. Standard schematic.
• A side-by-side of the two spellings of the ZX-plane with the same point marked on it. Standard schematic, and a thirty-second aside rather than a full panel.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 11 "Introduction to Three Dimensional Geometry", the boxed Note in §11.3, p. 210 — the origin, points of the x-axis, points of the YZ-plane
• §11.3, pp. 209–210 — the identification of the three coordinates with distances from the three planes, which is what makes a zero readable
• Fig 11.3, §11.3, p. 210 — the lettered box whose corners supply every zero pattern
• Example 1, p. 211 — a corner of that box found by locating the vanishing measurement
• Exercise 11.1 q1, q2 and q4, p. 211
• Summary, p. 215 — the three axis forms, and the absence of a matching plane form