The named kinds — zero, unit, coinitial, collinear, equal, negative and free

What to assume they know

• Vectors, magnitude and direction, and Definition 1, from the first topic of this module
• A directed line segment's two ends, the one it leaves and the one it reaches, and the arrowhead that tells them apart
• Parallel lines, and the fact that two parallel lines need not be the same line
• Compass bearings, and reading a stated angle from a named direction
• Reading a scale bar on a drawing, from earlier schooling
• The idea of an inverse under an operation, informally

What they should be able to do

• Recite the chapter's seven named kinds and say which page defines each
• Explain why the zero vector is the awkward case for Definition 1, and quote the chapter's own two-sentence patch
• Recognise the unit vector notation, and say where in the chapter the formula for building one actually appears
• Distinguish a property that belongs to two vectors from a property that belongs only to a particular drawing of them
• State the chapter's definition of collinear and show why it does not require one line
• State the chapter's definition of equal and identify the clause that makes position irrelevant
• Build the negative of a stated vector and name the two separate places the chapter defines it
• Explain what the closing Remark on free vectors licenses, and what would break without it
• Classify a set of drawn arrows as collinear, equal or coinitial
• Decide the truth of a claim relating collinearity, magnitude and equality, and give a counterexample when it is false
• Draw a displacement from a stated distance and bearing, to a stated scale

Where it usually goes wrong

• "Collinear means the vectors lie on one line." The chapter's own definition says parallel to the same line, and Example 3's printed key calls three vectors collinear that are not all drawn on one line. The word is a false friend and it is the single most reliable source of lost marks in this section.
• "Collinear vectors point the same way." The definition closes by waiving both length and sense. A vector and its negative are collinear. That is item one of Exercise 10.1 Q5 and it is printed as a claim to be judged true.
• "Two vectors of the same length pointing the same way but drawn in different places are different vectors." They are equal. The definition waives where each one starts, and the Remark then says every vector in the chapter may be slid anyway.
• "The zero vector has no direction, so it breaks the definition of a vector." It does sit awkwardly with Definition 1, and the chapter knows: it offers, in the next sentence, a convention under which the zero vector is granted whichever direction suits. Teach both sentences and say which one the rest of the chapter relies on.
• "Coinitial is a relation between two vectors." It is a relation between two drawings. Once the Remark on free vectors has been read, any two vectors can be made coinitial by sliding one. The chapter only ever asks the question about a supplied figure.
• "A unit vector is a special vector I have to be given." Any nonzero vector has one, and the recipe is in this book — six pages later. Do not let a student leave this topic thinking the hat is only ever handed to them.
• "The negative of a vector is what you get by putting a minus sign in front of the letter, and that is a different idea from reversing the arrow." They are the same idea, defined twice: geometrically on Part II p. 341 and as the scalar multiple by minus one on Part II p. 346.
• "The little arrows in the labels on Fig 10.6 show which way the sides point." They are the vector accent and they all point right. The side directions are in the drawn arrowheads on the square itself.
• "Free vectors are one more kind, alongside the other six." They are not a seventh category sitting beside the first six; the Remark says all of the six are free, and that all vectors in this chapter are free. It is a statement about the whole chapter, not an entry in a list.

Questions to check understanding

• Given a drawing of several arrows, name which are collinear, which are equal and which are coinitial — the form of Example 3 and Exercise 10.1 Q4
• Judge a claim relating collinearity, magnitude and equality, and supply a counterexample where it fails — the form of Exercise 10.1 Q5
• Draw a displacement of stated length and bearing to a stated scale — the form of Example 1 and Exercise 10.1 Q1
• Produce a pair of unequal vectors matched in magnitude, and a pair matched in direction — the form of Exercise 10.2 Q2 and Q3
• State what the zero vector's direction is taken to be, and why the chapter needs a convention at all
• Write the negative of a named vector in the chapter's two-letter notation and say what changes and what does not
• Explain what the Remark on free vectors permits, and name one later argument in the chapter that would fail without it

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 shape of §10.3 (Part II p. 341). Six bold run-in headings in a single column — zero, unit, coinitial, collinear, equal, negative — followed by an italic Remark that introduces the seventh word, free, and then applies it to everything above. Read off the printed page: the six are set as run-in headings, not as a numbered list, so a student skimming sees six paragraphs and not six definitions. Section 1 should put them back into a list.
• The zero vector, and the patch under it (§10.3, Part II p. 341). Definition 1 on the previous page demanded magnitude and direction. The zero vector's own entry then says it cannot be assigned a definite direction because its magnitude is zero — and the very next sentence offers, as an alternative, that it may be granted whichever direction is convenient. The two sentences do not agree, and the chapter presents them as interchangeable. Verified as a reading, not as a printed claim: the chapter states both and reconciles neither. This is the sharpest thing on the page and section 2 exists for it. Say plainly that the second sentence is the convention the rest of the chapter uses, because every later result that multiplies or crosses the zero vector needs it.
• The zero vector's two examples, and the comma (Part II p. 341). The chapter names two directed line segments that stand for the zero vector — each one from a point back to itself. That sentence ends with a comma rather than a full stop, so it runs straight into the bold heading below it. Confirmed on the printed page. The first topic of this module records the same slip; it is recorded here too because this is the topic that shows the sentence.
• The unit vector entry (§10.3, Part II p. 341). Two lines: a vector of magnitude one is a unit vector, and the unit vector along a given vector is written with a hat over the letter. No formula is given. Verified as absent from §10.3 and present elsewhere: the construction — divide the vector by its own magnitude — is printed at Part II p. 347, six pages later, at the end of §10.5, and it is the second topic of module two that carries it. Section 3 should name the hat, promise the formula and hand it forward rather than invent it here.
• Coinitial vectors (§10.3, Part II p. 341). Several arrows that share a starting point. Verified as a reading: this is the only one of the seven that is a property of where the arrows have been drawn rather than of the vectors themselves — which is exactly what the Remark six lines further down says may be changed at will. The chapter never notices the tension. Section 4 should, and should resolve it the way the chapter's own exercises do: coinitial is a question you ask about a supplied figure, not about a pair of vectors in the abstract.
• Collinear vectors (§10.3, Part II p. 341). The definition asks only that the vectors be parallel to one line, and then adds, in a clause of its own, that neither their lengths nor the way they point makes any difference. Both halves are load-bearing and neither is what the word suggests. Verified against Example 3 on the facing page: the vectors the printed key calls collinear are not all drawn on one line — one of them sits on a parallel line some distance away — so the chapter means parallel, not lying on a single line. And the clause about the way they point is what lets a vector be collinear with its own negative. Section 5 is the most important ninety seconds in this topic.
• Equal vectors (§10.3, Part II p. 341). Same magnitude, same direction, and then a closing clause saying that where each of the two starts is beside the point. Verified: that clause is the whole content of the definition, because without it equality would collapse into coincidence, and the Remark two paragraphs later would be unstatable. Show it in isolation.
• Negative of a vector (§10.3, Part II p. 341). Same magnitude, opposite direction; the chapter's instance is a segment and the segment with its two letters swapped, written with a minus sign. The same object is defined a second time on Part II p. 346, as the case where the scalar multiplier is minus one, and there it is given a second name, additive inverse. Verified: the two definitions agree and the chapter never says they are the same definition. Section 7 should show both entries side by side.
• The Remark on free vectors (§10.3, Part II p. 341). Any of the vectors just defined may be moved parallel to itself without changing magnitude or direction; and the chapter's word for vectors of that kind is free; it then announces that free ones are all it will handle for the rest of the chapter. Verified as scope-setting, not as an aside: the triangle law on Part II p. 343 slides one vector so its tail meets another's head, and the proof of commutativity on Part II p. 345 copies two sides of a parallelogram onto the opposite two. Neither move is legal without this Remark. The word appears on exactly one page of the chapter — confirmed by grep across all thirty-nine extracted pages and on the page image of each — and it is doing work on at least six later pages.
• Example 1 and Fig 10.4 (Part II p. 341). A displacement of forty kilometres, thirty degrees west of south, drawn as one arrow from a marked origin. Read off the printed page: the drawing carries the four compass letters, the angle marked between the arrow and the southward direction, the distance written along the arrow, and a separate scale bar reading ten kilometres. The scale bar is the part students omit and it is the part that makes the drawing an answer rather than a sketch.
• Exercise 10.1 Q1 (Part II p. 342). The same distance, thirty degrees east of north. Verified, and the chapter does not say it: thirty degrees east of north and thirty degrees west of south are separated by exactly half a turn, and the two displacements have the same length, so the answer to Q1 is the negative of the arrow drawn in Example 1 — and Negative of a Vector is defined on the same page as Example 1. Section 12 should draw both on one compass rose. This is an added observation; treat it as an editorial call and see the note below.
• Example 3 and Fig 10.5 (Part II p. 342). Four arrows and a scale bar. Read off the printed page: two of the four leave a point on the right, one leaves a point on the left and runs up, and one runs leftward between the two points; each arrow carries tick marks giving its length against the scale bar. The printed key names three collinear, two equal and three coinitial. Verified against the printed page: the three called collinear are the three drawn parallel to one another, and one of them is on a different line from the other two; the two called equal have the same tick count and the same sense; the three called coinitial are the three drawn out of the right-hand point.
• Exercise 10.1 Q4 and Fig 10.6 (Part II p. 342). A square with one arrow along each side. Read off the printed page: the top side points right, the bottom side points left, and both vertical sides point downwards. Verified: the two downward sides have equal length and the same sense, so they are the equal pair; the top arrow and the left arrow start at the same corner, so they are coinitial; and the top and bottom arrows are parallel with opposite senses, so they are collinear and not equal — which is precisely what part (iii) asks for. Note for the illustrator: the little arrows in the four labels all point rightwards, because they are the vector accent, not the vector. Reading them as directions gives the wrong answer to every part.
• Exercise 10.1 Q5 (Part II p. 342). Four claims to judge. Verified in order: a vector is collinear with its own negative — true, because the definition says the way they point makes no difference; two collinear vectors always have equal magnitude — false, since the same definition also waives length; two vectors of the same magnitude are collinear — false, and any two arrows of equal length at an angle is a counterexample; two collinear vectors of the same magnitude are equal — false, because they may be opposite, which is the first claim again. All four turn on the same two words in one definition, and the fourth is the one that decides whether a student has understood the topic.
• Exercise 10.2 Q2 and Q3 (Part II p. 354). Produce a pair of unequal vectors matched in magnitude; then a pair matched in direction. Verified: each has infinitely many answers and neither has a unique one, which is the point — Q2 forces a student to break equality on direction and Q3 to break it on magnitude, and together they take the definition of equal apart from both sides. They sit in a later exercise but they are this topic's questions.
• The Summary, and what it leaves out (Part II pp. 373–375). Verified as absent: of the eleven Summary bullets, not one defines or lists any of the seven kinds. The word coinitial does appear in the Summary, but inside the bullet stating the parallelogram law, not as a definition; the word free appears nowhere in the Summary at all. Confirmed by grep over the three Summary pages and on the page image of each. A student revising from the Summary alone would never meet this topic. Say so at the close.

Figures to have open

• A redraw of Fig 10.4 (Part II p. 341): compass letters at four points, one arrow from the centre into the lower-left quadrant, the angle marked between that arrow and the southward direction, the distance written along the arrow, and the scale bar beside it. The chapter's own drawing.
• A redraw of Fig 10.5 (Part II p. 342) for section 9: four arrows in the printed arrangement, with tick marks preserved, because the tick marks are how the equal pair is identified.
• A redraw of Fig 10.6 (Part II p. 342) for section 10: a square with one arrow on each side, the two vertical sides both pointing down, the top pointing right and the bottom pointing left. Label the four with plain letters and no accent, so the label arrows cannot be mistaken for the side directions.
• A single reusable arrow for sections 4, 6 and 8, drawn once and then copied to three positions on the frame, so that "same vector, different place" is carried by one image rather than three.
• A four-row table for section 11, built with the repo's DataTable component, carrying the four claims, the verdicts and a counterexample sketch for each false one. The counterexamples are not in the book; the exercise supplies none.
• One compass rose for section 12 carrying both displacements at once. Not in the book; the chapter draws only the first of the two.
• No figure is needed for sections 2, 3 or 5 beyond the kit shapes named above.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 10 "Vector Algebra", §10.3 Types of Vectors, the six run-in headings and the closing Remark, Part II p. 341
• Definition 1, for the contrast drawn in section 2, §10.2, Part II p. 339
• Example 1 with Fig 10.4, Part II p. 341; Example 3 with Fig 10.5, Part II p. 342
• Exercise 10.1, questions 1, 4 and 5, with Fig 10.6, Part II p. 342
• The unit vector formula, §10.5, Part II p. 347; the negative as the scalar multiple by minus one, and the name additive inverse, §10.5, Part II p. 346
• Exercise 10.2, questions 2 and 3, Part II p. 354
• Summary, Part II pp. 373–375, cited for what it does not carry