Magnitude, the three angles with the axes, and the cosines and ratios they give

What to assume they know

• A right handed system of rectangular coordinates in space, from Class XI
• The distance of a point from the origin in three dimensions, from Class XI
• Cosine of an angle as the adjacent side over the hypotenuse in a right angled triangle
• The Pythagorean identity, and cosine values at the standard angles
• What it means for two triples of numbers to be proportional
• Vectors, magnitude, and the two ends of a directed line segment, from the previous topic

What they should be able to do

• Say what makes a vector a position vector, and what it is measured against
• Write the magnitude of a position vector from the coordinates of its terminal point
• Identify the three angles the chapter measures, and say which direction along each axis they are measured from
• Derive the cosine of each of those angles as a coordinate over the length, and say what the derivation assumes about where the point lies
• Recover the coordinates of the point as the length times each cosine
• State the one identity the three cosines satisfy and use it to find a missing cosine up to sign
• Distinguish direction ratios from direction cosines, and explain why their squares do not add to one
• Read the direction ratios of a vector straight off its components, and convert them to cosines
• Handle the equally inclined case, and produce both signs of the answer

Where it usually goes wrong

• "Direction ratios and direction cosines are two names for the same three numbers." They agree only when the vector already has length one. The Note on Part II p. 341 exists because this is the error students actually make, and the fastest cure is to square and add both triples.
• "Any three numbers can be direction cosines." They must satisfy one identity, so only two of the three are free and the third is fixed up to sign. That is what Exercise 10.4 Q3 is testing, and it is why that item can ask for an angle after giving only two.
• "Direction ratios are unique." As the chapter defines them on Part II p. 340 they are one specific triple, the length times each cosine. Later work with lines treats any triple proportional to that one as a set of direction ratios. Both usages are current; say which one is shown. See the note below.
• "The angles are measured from the axes." They are measured from the positive directions of the axes. Drop the word positive and every sign in Exercise 10.2 Q13 becomes arguable.
• "The triangle argument proves the formula." It proves it for a point whose coordinates are not negative. The cosine formula holds for every point, and the chapter's own general proof is on Part II p. 358, in a different section, under a different heading. An explanation that runs only the triangle has proved less than it claimed.
• "A negative direction cosine is a mistake." An obtuse angle with an axis gives a negative cosine, and Exercise 10.2 Q13 produces two of them. The quantity that can never be negative is the length, not the cosines.
• "Equally inclined means the angles are forty-five degrees." It means the three are equal to each other; the identity then fixes each cosine at plus or minus one over root three, which is not a standard angle at all.
• "Reversing the vector leaves the direction cosines alone." It negates all three, which is why Miscellaneous Exercise Q11 carries a plus-or-minus and Exercise 10.2 Q13 says which way round to travel.

Questions to check understanding

• Find the magnitude of a vector given the coordinates of its terminal point
• Write the direction ratios of a vector given in component form, and convert them to direction cosines — the form of Example 9 and Exercise 10.2 Q12
• Produce the three cosines for a vector running between two named points, travelled one stated way — the form of Exercise 10.2 Q13
• Show that a given vector is equally inclined to the three axes
• Given two of the three direction angles of a unit vector and a constraint on the third, find the third and the components — the form of Exercise 10.4 Q3
• Decide whether a stated triple can be a set of direction cosines
• Explain why the sum of the squares of the direction ratios is not one in general, and say when it is

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 position vector paragraph and Fig 10.2 (§10.2, Part II pp. 339–340). A point P in space with coordinates x, y and z is joined to the origin, and the resulting vector is named the position vector of P. Its magnitude is written as the square root of the sum of the three squared coordinates, taken over from the Class XI distance formula rather than derived here. Read off the page image, Fig 10.2 has two panels: (i) is a bare set of three axes with one arrow drawn from the origin to a labelled point; (ii) shows three arrows leaving one origin and reaching three different labelled points, which is the chapter's picture for writing several position vectors with single letters at once.
• Fig 10.3, and the inset beside it (Part II p. 340). The main drawing is a rectangular box with the origin at one corner and the point at the opposite one, the three edge feet marked on the three axes, the vector drawn as the long diagonal, and the three angles marked at the origin. To its right the page prints a second, smaller drawing — the same origin, the same point, one axis foot, and a square right angle mark. That inset is the whole derivation of section 4 and is easy to miss at a glance.
• The three cosine formulas (§10.2, Part II p. 340). Each cosine is a coordinate divided by the length. Verified, and with the assumption named: the argument the chapter gives is that the triangle formed by the origin, the foot on an axis, and the point is right angled at the foot, so the cosine of the angle at the origin is the adjacent edge over the hypotenuse. That reads the adjacent edge as a positive length, which is only the coordinate itself when the coordinate is not negative. The formulas are nonetheless right for every point, and the chapter proves them again for every point on Part II p. 358 as a scalar product — see the last bullet. Section 4 should name the gap and section 11 should close it.
• Reading the coordinates back (§10.2, Part II p. 340). Since each coordinate is the length times a cosine, the point can be written as the length times the triple of cosines. The chapter then calls that same triple of products the direction ratios and gives them the letters a, b and c. Verified: the ratios are therefore the coordinates of the point over again, which is exactly what Remark (ii) on Part II p. 349 says when the vector arrives in component form instead.
• The boxed Note on Part II p. 341. The three cosines squared add to one; the three ratios squared do not, in general. Verified: squaring and adding the three cosine formulas gives the sum of the squared coordinates over the squared length, which is one by the magnitude formula on the previous page. Squaring and adding the ratios gives the squared length itself, which is one only when the vector already had length one. Show both computations; the Note asserts the two facts and derives neither.
• Example 9 (Part II p. 351). The vector with components one, one and minus two. The solution states that the ratios are just the components, giving one, one and minus two, then computes the length as the square root of six and divides. Verified: the length is indeed the square root of six, and the three cosines are one over root six, one over root six, and minus two over root six. Check the identity: one sixth plus one sixth plus four sixths is one.
• Remarks (ii) and (iii) (§10.5.1, Part II p. 349). Remark (ii) says the three components of a vector are also called its direction ratios. Remark (iii) says that the triple of direction cosines, assembled into a vector, is the unit vector in that direction. Verified: the second is the first Note of this topic restated — a triple whose squares sum to one has length one — and it is the cleanest single sentence in the chapter for tying this topic to the next module.
• Exercise 10.2 Q12 (Part II p. 354). A vector whose three components are one, two and three; find its cosines. Verified: the length is the square root of fourteen, so the cosines are one, two and three each over the square root of fourteen, and their squares add to fourteen over fourteen.
• Exercise 10.2 Q13 (Part II p. 354). A vector that runs from the point with coordinates one, two and minus three to the point with coordinates minus one, minus two and one, travelled in that order; find its cosines. Verified: the joining vector has components minus two, minus four and four; its length is six; the cosines are minus one third, minus two thirds and two thirds. This is the item that most rewards a slow read, because the direction of travel decides all three signs and reversing it flips all three.
• Exercise 10.2 Q14 (Part II p. 354). Show that the vector with all three components equal to one is equally inclined to the three axes. Verified: the length is the square root of three and every cosine is one over the square root of three, so the three angles are equal. Their common value is the angle whose cosine is one over root three, which the chapter does not ask for.
• Exercise 10.4 Q3 (Part II p. 368). A unit vector makes a third of a half-turn with the first axis vector, a quarter of a half-turn with the second, and an acute angle with the third; find that angle and the components. Verified: the two known cosines are one half and one over root two, whose squares are a quarter and a half, leaving a quarter for the third square, so the third cosine is plus or minus one half and the acute condition picks the plus. The angle is a third of a half-turn and the components are one half, one over root two, one half. The item is printed inside Exercise 10.4, which is otherwise entirely about the vector product; it belongs to this topic and the explanation for the last topic should hand it back here.
• Miscellaneous Exercise Q11 (Part II p. 372). Show that a vector equally inclined to all three axes has all three cosines equal to plus or minus one over the square root of three. Verified: equal angles force equal cosines, and the identity then forces three times the common square to be one. Both signs are genuine — the printed statement carries the plus-or-minus.
• The Remark on Part II p. 358. In the projection section, the chapter computes the same three cosines again for a vector given in component form. Read off the printed page, because the text layer drops the whole display: the first of the three is written twice over — once as a scalar product, taking the vector against the first axis unit vector and dividing by the product of the two lengths, and then reduced to the first component over the length. The second and third are printed only in the reduced form; the chapter does not write the scalar product out for them. Verified: the one derivation that is written out uses no triangle and no assumption about signs, and the other two follow by the same route, so between them they cover every point in space. This is the honest general proof of what Part II p. 340 asserted from one picture, and section 11 is worth ninety seconds for exactly that reason. Say that the book writes it once and leaves the other two to the reader.
• The Summary bullets (Part II p. 373). Three of the eleven belong here: the position vector with its magnitude; the sentence identifying scalar components with direction ratios and calling them projections along the axes; and the three quotients tying length, ratios and cosines together. Note that the middle one compresses two facts printed nine pages apart in the running text — the identification with the components is Remark (ii) on Part II p. 349, and the identification with the projections is the second half of the Remark on Part II p. 358. The Summary joins them; no running page does.

Figures to have open

• A redraw of Fig 10.3's main box (Part II p. 340): three axes, the point at the far corner of a rectangular box, the three feet marked on the three axes, the vector as the long diagonal, and the three angles marked at the origin. The chapter's own drawing.
• A large redraw of the small inset printed to the right of Fig 10.3 on the same page: the origin, the point, one axis foot, and the square right angle mark. The chapter draws it small and unlabelled beyond the right angle; section 4 cannot be told without it.
• A single reusable diagram for sections 5 and 6 in which the three cosines and the length are shown as four labels on one arrow, so that the products and the identity can both be built on top of it without redrawing.
• A two-column comparison for section 7, cosines against ratios, built with the repo's Compare component, with the scaling factor shown once between them.
• No figure is needed for sections 8 to 11; those are algebra, and the chapter supplies none for any of it. Confirmed on the page image of each page: Part II pp. 349 and 350 carry no figure at all, and neither Remark (ii) or (iii), nor Example 9 on Part II p. 351, nor Exercise 10.2 on Part II p. 354 has one attached. Part II pp. 351, 352 and 353 do each print a figure — Fig 10.15, Fig 10.16 and Fig 10.17 — but all three belong to the section formula and to the vector joining two points, not to this topic.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 10 "Vector Algebra", §10.2, Position Vector, Part II p. 339; Fig 10.2, which is called on that page but printed at the head of Part II p. 340
• §10.2, Direction Cosines, Part II p. 340, with Fig 10.3 and its inset, and the boxed Note at the head of Part II p. 341
• §10.5.1 Remarks (ii) and (iii), Part II p. 349
• Example 9, Part II p. 351
• Exercise 10.2, questions 12, 13 and 14, Part II p. 354
• Exercise 10.4, question 3, Part II p. 368; Miscellaneous Exercise, question 11, Part II p. 372
• The Remark deriving the same three cosines as scalar products, Part II p. 358; Summary, first three bullets, Part II p. 373