Direction ratios as any proportional triple, and normalising back to cosines

What to assume they know

• Direction angles and direction cosines, and the identity that their squares total one — the previous topic
• Proportional triples, and what it means for two triples to be in the same ratio
• Square roots, and the fact that a square root has two signs
• Subtracting coordinates to get the gaps between two points in space
• Collinear points, from Class XI coordinate geometry
• Cancelling a common non-zero factor from a ratio

What they should be able to do

• Define a direction-ratio triple as any triple proportional to the direction cosines, and state the scaling relation the chapter writes
• Explain why one line has endlessly many such triples while it has at most two cosine triples
• Recover the direction cosines from a given ratio triple by dividing through by the length of the triple
• Derive that recovery, rather than quoting it, from the identity of the previous topic
• Account for the plus-or-minus in front of the recovered cosines, and say what a choice of sign chooses
• Take direction ratios straight off two points, in either subtraction order, and explain why both orders are legitimate
• Test three points for collinearity using proportional ratios plus a shared point, and say why the shared point is not optional
• Recognise the alternative name the chapter reports some writers use
• Distinguish a ratio triple from a cosine triple in a context where confusing them changes an answer

Where it usually goes wrong

• "Direction ratios are just direction cosines that have not been simplified." They are a different kind of object. Cosines are a specific triple with a fixed length; ratios are an entire family, closed under scaling by any non-zero number, and no member of that family is privileged. The chapter's own remark that there are endlessly many is the point, not an aside.
• "So I can scale the entries individually to make them tidy." One multiplier has to serve all three entries at once. Doubling the first and tripling the second gives a triple for a different line.
• "Dividing by the root always gives the right answer." It gives one of the two right answers. The chapter prints a plus-or-minus, and the Summary drops it. A question that fixes a sense — an angle stated as obtuse, or a specified direction of travel — decides which one.
• "Proportional ratios mean the points are collinear." They mean the segments are parallel. Two parallel segments in space that share no point are two segments, not one line. Example 5 says so in a sentence students routinely skim.
• "The denominator in a normalisation is the distance between the points." It is the length of whichever triple you were handed, and the triple need not be the coordinate gaps at all. Example 2 normalises a triple that came from nowhere in particular; Exercise 11.1 Q3 does the same. Only when the ratios happen to be the coordinate gaps does the root also mean a distance.
• "Reversing the subtraction order is an error to be corrected." The chapter's own Note offers both orders. They differ by an overall sign, which is a rescaling, and both describe the same line.
• "A ratio triple can be zero." Not all three at once — the chapter's multiplier is required to be non-zero, and a triple of three zeros has no root to divide by. Individual entries may certainly be zero, and one of them being zero says the line is perpendicular to that axis.
• "Direction numbers is a different concept." It is the same triple under another name, and this chapter says so in a boxed line.

Questions to check understanding

• Given a ratio triple, produce both cosine triples and check the identity on one of them — the form of Exercise 11.1 Q3
• Given a cosine triple, produce three different ratio triples for the same line
• Show that three named points lie on one line, stating both halves of the argument — the form of Example 5 and of Exercise 11.1 Q4
• Given two points, write ratio triples in both subtraction orders and say how they are related
• Decide whether a stated triple can be a cosine triple, a ratio triple, both, or neither
• Explain why a line has two cosine triples but endlessly many ratio triples
• Given a ratio triple with a zero entry, say what that zero tells you about the line's position relative to the axes

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The definition (§11.2, Part II p. 378). The chapter defines the triple by proportionality to the cosines and then writes the relation out with a single non-zero real multiplier applied to each of the three cosines in turn.
• The boxed Note (Part II p. 378). One line reporting that some authors use a different name for the same triple. Worth ten seconds: a student meeting the other name in a reference book should not think it is a new object.
• The recovery derivation (Part II pp. 378–379). Verified, and this is the spine of the topic: set each cosine over its matching ratio equal to a common constant, so each cosine is that constant times its ratio; square and add the three; the left-hand side is one by the identity of the previous topic, so the square of the constant times the summed squares of the ratios is one; hence the constant is plus or minus the reciprocal of the square root of the summed squares. Substituting back gives each cosine as its ratio over that root. The chapter runs exactly this argument across the page break, and it is the only place in the chapter where the identity does any work. Run it as an argument, not as a formula to be memorised.
• The plus-or-minus (Part II p. 379). The chapter is explicit that the sign follows the sign wanted for the constant, and that one sign or the other is taken across all three entries together. Verified: the two choices are the two directions along the line, which is the two-sets fact from the previous topic arriving in algebraic clothing. A ratio triple does not carry a sense; it carries a line.
• Endlessly many triples (Part II p. 379). Multiplying a ratio triple by any non-zero number gives another ratio triple, so a line has infinitely many, and any two of them stand in a fixed ratio to each other. Verified: the recovery above kills the common factor, top and bottom, so all of them normalise to the same pair of cosine triples. That invariance is what makes the ratios usable.
• Example 2 (Part II p. 380). Ratios two, minus one, minus two. Verified: the summed squares are four plus one plus four, which is nine, whose root is three, so the cosines are two thirds, minus one third and minus two thirds. Check: four ninths plus one ninth plus four ninths is one. The chapter writes the denominator out three times, unsimplified, before giving the tidy triple.
• The Note on a segment's ratios (Part II p. 380). For the segment between two named points, the ratios may be taken as the three coordinate gaps — and the Note prints both subtraction orders as acceptable. Verified: the two orders differ by a factor of minus one throughout, which is a legitimate rescaling, so both are ratio triples for the same line; they normalise to the two opposite cosine triples. This is the clearest single illustration in the chapter that a ratio triple does not fix a sense.
• Example 5 (Part II p. 381). Three points; the first two give one ratio triple and the second two give another. Verified: from the first point to the second the gaps are minus one, minus five, seven; from the second to the third they are two, ten, minus fourteen, which is exactly minus two times the first triple. Proportional, and the middle point lies on both segments, so the three points are collinear. The chapter's argument has two halves and students drop the second. Proportionality alone gives two parallel segments; it is the shared point that fuses them into one line.
• Exercise 11.1 Q3 (Part II p. 381). Ratios minus eighteen, twelve, minus four. Verified: the summed squares are three hundred twenty-four plus one hundred forty-four plus sixteen, which is four hundred eighty-four, and the root is twenty-two exactly. Dividing gives minus nine over eleven, six over eleven and minus two over eleven. Check: eighty-one plus thirty-six plus four is one hundred twenty-one, over one hundred twenty-one. This is the tidiest normalisation in the chapter and should be the one worked.
• Exercise 11.1 Q4 (Part II p. 381). Three points to be shown collinear. Verified: from the first point to the second the gaps are minus three, minus five, minus three; from the first to the third they are three, five, three, the exact negatives. Proportional with the first point common, so the three are collinear. Note that this item can be settled from either pair, which is a good thirty seconds on why the choice of pair does not matter.
• The Summary bullets (Part II p. 391). Two bullets belong here: one defining the ratios by proportionality, and one giving each cosine as its ratio over the root of the summed squares. Verified against the body: the Summary prints the recovery without the plus-or-minus that Part II p. 379 prints. A student revising from the Summary alone loses the sign choice, and with it the fact that a line has two senses. Read closely on the printed page and unambiguous; make it a beat of its own.

Figures to have open

• A single line drawn once, with four ratio triples arranged around it and arrows running from all four to one cosine triple. Not in the book; the chapter states the fact and draws nothing. Carries section 3 and should reappear behind section 12.
• The same line with two arrowheads and the two opposite cosine triples attached, one to each. Not in the book, for section 6. It is the picture from the previous topic's section 3, reused deliberately so the two topics visibly connect.
• Two points in an axis frame with the three coordinate gaps drawn as three axis-parallel steps, then the same picture with the steps reversed. Not in the book, for section 8; the chapter prints this material as a boxed Note with no figure.
• A schematic for section 9 showing two parallel segments that share no point, beside two that do. Not in the book, and the whole content of the collinearity misconception.
• No figure is needed for sections 2, 5, 7, 10 or 11; those are algebra and arithmetic.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 11 "Three Dimensional Geometry", §11.2, the definition of the ratios and the boxed Note on the alternative name, Part II p. 378
• The recovery of the cosines from the ratios, across the page break, Part II pp. 378–379
• The boxed Note on a segment's ratios in both orders, Part II p. 380
• Examples 2 and 5, Part II pp. 380–381; Exercise 11.1 questions 3 and 4, Part II p. 381
• Summary, the fourth and fifth bullets, Part II p. 391