The angle between two lines, from cosines or from ratios

What to assume they know

• The dot product of two vectors, and its expression as a product of lengths times a cosine, from Chapter 10 of this volume
• Direction cosines, direction ratios, and the identity that the cosines square to one — the first module
• Reading a direction-ratio triple off a vector equation or a symmetric Cartesian equation — the second module
• The sign of the cosine on either side of a right angle, and the modulus of a real number
• The identity relating the square of a sine to the square of a cosine
• Solving a linear equation for one unknown

What they should be able to do

• Explain why two lines that never meet still have a well-defined angle, and what construction supplies it
• Derive the cosine formula from the dot product, using direction ratios as components
• Say what the modulus bars around the whole quotient do, and why the chapter puts them there
• Write the same formula for normalised triples, and account for the denominator disappearing
• State the perpendicularity condition and the parallelism condition, and say which of the two formulas each comes from
• Apply the formula to two lines given in vector form and to two given in Cartesian form
• Solve for an unknown appearing in a denominator so that two lines meet at right angles
• Recognise a pair of lines that are perpendicular for reasons that survive any choice of numbers
• Identify the one printed sine formula in this section that cannot be right, and say how it should read

Where it usually goes wrong

• "Two lines that never meet have no angle." They have one, supplied by the Note on Part II p. 384: slide each to a parallel through a common point and measure there. Without that construction the whole section would apply only to intersecting lines, and every exercise item in it would be ill-posed.
• "The formula gives the angle." It gives the acute one. The modulus bars throw away the sign, and the sign was the only record of which of the two directions along each line had been chosen. If a question wants the obtuse angle, subtract from a straight angle afterwards.
• "A negative value means I made an arithmetic error." Before the bars are applied it means the two triples were pointing away from each other. The bars are in the chapter's formula precisely so this never reaches the answer.
• "Perpendicular needs the whole formula." It needs the numerator only. The denominator is a product of two lengths and can never be zero for genuine direction triples, so the quotient vanishes exactly when the sum of the three products does. Miscellaneous Exercise Q1 is a whole question resting on this.
• "Parallel means the ratios are equal." It means they agree in ratio, term by term — one triple is a fixed multiple of the other. Exercise 11.2 Q3 has a multiple of minus one, so the entries are not equal and the lines are still parallel.
• "I can read the ratios straight off any printed equation." Not if a numerator carries a coefficient, and not if the running variable has been subtracted from the constant rather than the other way round. Exercise 11.2 Q10 has both problems in one item, and it is the reason that item is hard.
• "The sine formula is a second, independent result." It is derived from the cosine formula on the same page, in three printed lines, and its only use in the chapter is to supply the parallelism condition. Do not memorise it; the printed version for cosines is in any case misprinted.
• "Skew lines are defined in this section." They are not; the word first appears in the next section. §11.4 speaks only of two lines, and its Note quietly covers the skew case without naming it.

Questions to check understanding

• Compute the angle for two lines stated in vector form — the form of Example 7 and of Exercise 11.2 Q8
• Compute the angle for two lines stated in symmetric Cartesian form — the form of Example 8 and of Exercise 11.2 Q9
• Show two lines perpendicular from two pairs of points — the form of Exercise 11.2 Q2
• Show two lines parallel from two pairs of points — the form of Exercise 11.2 Q3
• Show three given cosine triples mutually perpendicular — the form of Exercise 11.2 Q1
• Solve for an unknown in the denominators so that two lines meet at right angles — the form of Exercise 11.2 Q10 and of Miscellaneous Exercise Q3
• Prove a perpendicularity that holds for every value of the letters involved — the form of Miscellaneous Exercise Q1
• Explain what construction gives an angle to two lines that never meet

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 setup and Fig 11.4 (§11.4, Part II p. 383). The section opens with both lines already passing through the origin, each carrying a direction-ratio triple, with a point marked on each. Read off the printed page, the figure sets the origin at the left with the three axes, draws both lines through the origin with arrowheads at both ends, marks a point on each with a dot, labels the two lines at their upper-right ends, and marks the angle at the origin between the two forward rays. The angle is marked only between the two forward rays — the backward rays are drawn and carry no mark. That is the whole reason the chapter can call the answer acute.
• The Note that generalises it (Part II p. 384). Should either line miss the origin, the chapter replaces it by a parallel line that does not. Verified: parallel lines share direction ratios, which is the first module's Remark arriving where it is needed. This one boxed note is what makes the section apply to two lines anywhere in space, including two that never meet, and it is three lines long.
• The derivation of the cosine formula (§11.4, Part II p. 383). The two directed segments from the origin are vectors whose components are the two ratio triples, so the angle between them comes straight from the dot product: the sum of the three products of matching entries, over the product of the two lengths. Verified: nothing is used here beyond Chapter 10's dot product; the only new idea is reading a direction-ratio triple as a set of components, and that came out of the second module's Remark.
• The modulus bars (Part II pp. 383–384). The chapter writes the quotient inside modulus bars in every version of the formula it prints in the body. Verified as necessary: a line has two directions, so the raw quotient can come out negative depending on which triples were chosen, and the two answers are supplementary. The bars discard that choice and always return the acute one. The chapter puts the bars in and never explains them. Section 4 exists for this.
• The normalised version (Part II p. 384). With direction cosines in place of ratios, the formula reduces to the modulus of the sum of the three products, and the chapter prints the reason in a bracket beside it: each triple already squares to one. Verified: the denominator becomes one times one. This is the first module's identity doing its second piece of work in the chapter.
• The sine formula, and a printed sign error (Part II pp. 383–384). The chapter gives a companion formula for the sine, twice: once for ratios, numbered two on Part II p. 383, and once for cosines, numbered four on Part II p. 384. The ratio version is derived in three printed lines from the cosine version and comes out with three squared brackets added together under the root. The cosine version prints a minus between the first and second squared brackets where a plus belongs. Read on the printed page and unambiguous. Verified as an error: cosines are ratios that happen to be normalised, so the second formula must be the first with the denominators set to one, and the first has all three signs positive.
• The two conditions (Part II p. 384). Perpendicular when the sum of the three products of matching entries is zero; parallel when the three ratios of matching entries agree. Verified: the first is the numerator of the cosine formula vanishing, and the chapter attributes it to that formula; the second is the sine vanishing, and the chapter attributes it to the sine formula. The chapter labels which formula each comes. — it is the only place the sine formula is used for anything.
• The vector and Cartesian restatements (Part II p. 384). The same formula written for two lines given by vector equations, using the dot product of the two direction vectors over the product of their lengths; and for two given in symmetric Cartesian form, using the ratios read off the denominators. Both are restatements, not new results.
• Example 7 (Part II pp. 384–385). Two lines in vector form with directions one, two, two and three, two, six. Verified: the numerator is three plus four plus twelve, which is nineteen; the two lengths are three and seven; so the cosine is nineteen over twenty-one. Note that the lengths come out whole, which is why this pair was chosen.
• Example 8 (Part II p. 385). Two lines in Cartesian form with ratios three, five, four and one, one, two. Verified: the numerator is three plus five plus eight, which is sixteen; the lengths are the roots of fifty and of six; sixteen over the root of three hundred simplifies to eight root three over fifteen, which is what the chapter prints. Work the simplification — it is two steps and students skip it.
• Exercise 11.2 Q1 (Part II p. 389). Three triples of direction cosines, to be shown mutually perpendicular. Verified: each triple has entries over thirteen whose squares total one hundred sixty-nine, so all three are genuine cosine triples; and all three pairwise sums of products come to zero. Three checks, each two lines. A good item because the identity and the perpendicularity condition are both exercised on the same numbers.
• Exercise 11.2 Q2 and Q3 (Part II p. 389). Ratios taken from pairs of points. Verified: in Q2 the two triples are two, five, minus four and three, two, four, whose products sum to six plus ten minus sixteen, which is zero — perpendicular. In Q3 the two triples are minus two, minus four, minus four and two, four, four, which are exact negatives, so the ratios agree and the lines are parallel. These two items are the bridge from the first module: the ratios come from subtracting coordinates, and nothing else is new.
• Exercise 11.2 Q8 (Part II pp. 389–390). Two pairs of lines in vector form. Verified: part one has directions three, two, six and one, two, two, giving a numerator of nineteen over lengths seven and three — the same arithmetic as Example 7 with the two lines exchanged, and the same answer, nineteen over twenty-one. Part two has directions one, minus one, minus two and three, minus five, minus four, giving a numerator of sixteen over the roots of six and of fifty, which reduces to eight over five root three. Worth flagging that part one duplicates the worked example exactly; a student who notices has understood that the order of the two lines does not matter.
• Exercise 11.2 Q9 (Part II p. 390). Two pairs in Cartesian form. Verified: part one has ratios two, five, minus three and minus one, eight, four, giving twenty-six over nine root thirty-eight. Part two has ratios two, two, one and four, one, eight, giving eighteen over twenty-seven, which is two thirds — the cleanest angle in the chapter, and the one to show.
• Exercise 11.2 Q10 (Part II p. 390). An unknown sits in two denominators, and the lines are to be made perpendicular. Verified: both printed equations need rearranging before the ratios can be read, because several numerators carry a coefficient or a reversed sign. Rewriting the first line gives ratios minus three, two-sevenths of the unknown, and two; the second gives minus three-sevenths of the unknown, one, and minus five. Setting the sum of products to zero gives eleven sevenths of the unknown equal to ten, so the unknown is seventy over eleven. This is the hardest item in the chapter and almost all of the difficulty is in the rearranging, not in the perpendicularity. Give it a section of its own.
• Exercise 11.2 Q11 (Part II p. 390). Two lines with ratios seven, minus five, one and one, two, three. Verified: seven minus ten plus three is zero. One line of arithmetic; use it as the warm-up before Q10.
• Miscellaneous Exercise Q1 (Part II p. 390). Two triples built from three letters: the first is the three letters themselves, the second is each letter minus the next in a cycle. Verified: the sum of the three products expands to six terms which cancel in pairs, leaving zero, so the angle is a right angle for every choice of the three letters. This is the best item in the chapter for an explanation, because it shows the condition doing something no arithmetic could — no numbers are ever substituted.
• Miscellaneous Exercise Q3 (Part II p. 391). An unknown appears in one denominator of each line, and perpendicularity is imposed. Verified: the ratios are minus three, twice the unknown, two and three times the unknown, one, minus five; the sum of products is minus nine times the unknown, plus twice the unknown, minus ten; setting it to zero gives the unknown as minus ten sevenths. Simpler than Q10 because neither equation needs rearranging.
• The Summary bullets (Part II pp. 391–392). Four belong here: the cosine formula for cosines, the cosine formula for ratios, the vector version, and the Cartesian version with cosines beneath the fractions. There is a fifth bullet defining the angle between skew lines — see the note below, because that definition is not in the body.

Figures to have open

• Two plainly non-meeting lines in an axis frame for section 1. Not in the book; the chapter draws nothing for the Note that handles this case, and Fig 11.4 shows both lines already through the origin.
• The translation movement for section 2: the same two lines, each acquiring a parallel copy through one point, the copies meeting. Not in the book, and the most important new drawing in this brief.
• A redraw of Fig 11.4 (Part II p. 383) for section 3: both lines through the origin with arrowheads at both ends, a marked point on each, and the angle marked only between the two forward rays, as the printed figure marks it. Add the two component triples as labels, which the printed figure does not carry.
• A side-by-side slide for section 6 with the printed sine formula above and the corrected one below, the changed sign highlighted. Not in the book.
• No figure is needed for sections 4, 5, 7, 8, 9, 10, 11 or 12; those are notation and arithmetic. Note that peers laid out side by side in sections 5 and the worked examples must share one type size across all panels.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 11 "Three Dimensional Geometry", §11.4 Angle between Two Lines, the setup with Fig 11.4 and the cosine and sine formulas, Part II p. 383
• The boxed Note on lines not through the origin, the normalised formulas, the perpendicularity and parallelism conditions, and the vector and Cartesian restatements, Part II p. 384
• Examples 7 and 8, Part II pp. 384–385
• Exercise 11.2 questions 1, 2, 3, 8, 9, 10 and 11, Part II pp. 389–390
• Miscellaneous Exercise questions 1 and 3, Part II pp. 390–391
• Summary, the four angle bullets and the skew-angle bullet, Part II pp. 391–392