Three angles with the axes, and why their cosines square up to one

What to assume they know

• Rectangular axes in space, the coordinates of a point, and the eight regions the three coordinate planes cut space into, from Class XI
• The distance between two points in space, from Class XI
• Cosine of an angle between zero and one hundred eighty degrees, and its sign on each side of a right angle
• The cosine of a supplement, from Class XI trigonometry
• A right-angled triangle, and the cosine as adjacent over hypotenuse
• Chapter 10 of this book, which introduces the direction angles and their cosines for a vector before this chapter recalls them

What they should be able to do

• Draw a directed line through the origin and mark the three angles it makes with the positive axes
• Define the three direction cosines as the cosines of those three angles
• Explain why reversing the direction of the line replaces each angle by its supplement and flips the sign of all three cosines together
• State why an undirected line in space carries two sets of these numbers, and what choosing a direction buys
• Apply the chapter's remark for a line that misses the origin, by translating it to a parallel line through the origin
• State the identity that the three squares add to one, and locate the exact place the chapter uses it without having established it
• Derive that identity from the chapter's own two-point expression
• Reproduce the two-point derivation from the right-angled triangle in the chapter's figure, including the step the prose leaves to the picture
• Compute direction cosines from three stated angles, and from two stated points
• Write down the direction cosines of each coordinate axis and check the identity on all three
• Recognise which parts of this chapter's opening announcement are not delivered anywhere in the chapter

Where it usually goes wrong

• "The three direction angles are independent, so I can pick any three." They are pinned together by the identity: their cosines must square to one. Two of them, plus a sign, already determine the third. A student who has not met the identity as a constraint treats it as a formula to be recalled rather than as the thing that makes three angles describe one direction.
• "Equal angles with the three axes means each cosine is one third." It means three times the common square is one, so each cosine is one over root three, about zero point five seven seven, and the angle is about fifty-four degrees. Exercise 11.1 Q2 is where this error surfaces every year.
• "Reversing the line flips one of the signs." It flips all three, because all three angles turn into their own supplements at the same moment. There is no way to reverse a line halfway.
• "A direction cosine can be bigger than one." It is a cosine, so it lies between minus one and one, and the identity makes an individual value of one force the other two to zero — which happens exactly along an axis. That is Example 4.
• "A line not through the origin has no direction cosines." The Remark on Part II p. 378 slides a parallel copy to the origin and reads them there. This is the chapter's one sentence on the point and it is easy to miss.
• "The angle in the triangle is gamma because it looks like it." It is gamma because the short vertical stub in Fig 11.2 (b) is parallel to the z-axis, and the segment cuts two parallels. The chapter draws the argument and does not write it. An explanation that copies the assertion without the parallel has copied the gap.
• "A negative direction cosine is a mistake." An obtuse direction angle gives a negative cosine, and Exercise 11.1 Q1 has one. Signs carry the sense of the line; stripping them throws that away.
• "Direction cosines belong to a segment, so they change if I extend it." They belong to the direction, and every point of the line further out gives the same three numbers, because both the gaps and the length scale together. The chapter's own §11.2.1 phrasing says segment, which invites the confusion.
• "The chapter will get to planes later." It will not. This is the largest single thing an explanation can get wrong about this chapter, and it is planted by the chapter's own opening page.

Questions to check understanding

• Given three angles, produce the three direction cosines and verify the identity
• Given a triple, decide whether it can be a direction-cosine triple at all
• State what happens to all three cosines when the sense of a line is reversed, and justify it from the cosine of a supplement
• Produce the three cosines for the line through two named points — the form of Example 3 and of Exercise 11.1 Q5
• Work out the three cosines for a line opening equal angles with all three axes, keeping both signs — the form of Exercise 11.1 Q2
• Write down the direction cosines of a named coordinate axis and check the identity — the form of Example 4
• Derive the identity from the two-point expression, given the two points
• Explain how to obtain the three cosines for a line that misses the origin entirely

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 opening recall (§11.2, Part II p. 377). The chapter does not define the direction angles here — it points back at Chapter 10 of the same volume and recalls them: a directed line through the origin, three angles against the three axes, and the cosines of those three angles. Not verified: the Chapter 10 pack was not built.
• The supplement sentence (§11.2, Part II p. 377). Reversing the sense of the line replaces each of the three angles by its supplement, and every one of the three cosines changes sign. Verified: the cosine of a supplement is the negative of the cosine, so the three signs cannot flip independently — they go together. That "together" is the load-bearing word and the chapter does not stress it.
• Fig 11.1 (Part II p. 378). Read off the printed page: the z-axis runs up with an arrowhead, the y-axis to the right, the x-axis forward-left, and the origin is labelled O. A single straight line carries an arrowhead at its far end and is labelled L there; a point P is marked partway along it, and the segment from the origin out to P is labelled r. A dashed box shows the coordinate projections, its three edges at the origin labelled x, y and z. Three arcs sit at the origin: gamma against the z-axis, beta against the y-axis and alpha against the x-axis. The line is drawn only outward from the origin — it is not continued backwards through it. Note: the figure labels the segment r, a symbol §11.2 never introduces; r arrives later, in §11.3, as a position vector. Do not show the letter in this topic.
• Two sets, one set (Part II p. 378). A line in space runs both ways, so it offers two triples; committing to a direction picks one, and only then are the three numbers written l, m and n. Verified: the two triples are negatives of each other, by the supplement sentence on the previous page.
• The Remark for a line off the origin (Part II p. 378). Draw the parallel through the origin and read the numbers there, because parallel lines share them. This is the chapter's whole answer to a very common student question and it is three lines long.
• The identity, and where it is spent (Part II p. 378). Midway through the direction-ratio derivation the chapter writes, with no preamble beyond the word But, that the three squares add to one, and immediately uses it to solve for a scale factor. It is nowhere derived and nowhere stated as a result of its own in the body. It reappears as the second bullet of the Summary (Part II p. 391) and as a parenthetical on Part II p. 384. Checked against the page image of every one of the seventeen folios.
• Paying for the identity out of §11.2.1 (Part II pp. 379–380). Verified, and this is the explanation's central move: the chapter's own expression for the direction cosines of the segment from one point to another puts each coordinate gap over the length of the segment, and the length is the square root of the summed squares of exactly those three gaps. Square the three quotients and add: the numerator becomes the sum of the squared gaps and the denominator is that same sum, so the total is one. Every ingredient is printed in this chapter, one page after the identity was already spent. The chapter never assembles them.
• §11.2.1 and Fig 11.2 (Part II p. 379). Two points are named with their coordinates. Perpendiculars are dropped from both onto the XY-plane, meeting it at two further points; then a perpendicular is drawn from the first point onto that dropped segment, meeting at a fourth. In the resulting right-angled triangle the chapter asserts that the angle at the upper point equals gamma, and reads the cosine as the vertical gap over the length of the segment. The two remaining cosines are given by symmetry, with no separate argument.
• The step the prose leaves to the picture (Fig 11.2 (b), Part II p. 379). Read off the printed page: panel (b) draws the small triangle with the left vertex, the right vertex and the upper vertex, and it marks gamma twice — once at the upper vertex, inside the triangle, and once at the lower-left vertex, between the line and a short vertical stub drawn there. The stub is parallel to the z-axis. That second mark is the whole justification that the triangle's angle is the direction angle — two parallels cut by the segment — and the prose states the equality without arguing it. Also read off the printed page: there is no right-angle square at the foot vertex in either panel, though the prose calls the triangle right-angled, and panel (a) does not label the origin while panel (b) does.
• Example 1 (Part II p. 380). A line makes ninety, sixty and thirty degrees with the three positive axes. Verified: the cosines are zero, one half and root three over two; and zero plus a quarter plus three quarters is one, so the triple passes the identity. The chapter does not run that check. Run it — it is the first place the identity earns its keep.
• Example 3 (Part II p. 380). The line through the point with coordinates minus two, four, minus five and the point with coordinates one, two, three. Verified: the coordinate gaps are three, minus two and eight; the sum of their squares is nine plus four plus sixty-four, which is seventy-seven; so the three cosines are three, minus two and eight, each over the square root of seventy-seven. Check: nine plus four plus sixty-four over seventy-seven is one.
• Example 4 (Part II p. 381). The direction cosines of the three axes themselves. Verified: the x-axis opens zero degrees with itself and a right angle with each of the other two, giving one, zero, zero; the other two axes give zero, one, zero and zero, zero, one by the same reading. This is the cheapest possible check of the identity and the chapter states the second and third results without working them.
• Exercise 11.1 Q1 (Part II p. 381). Angles of ninety, one hundred thirty-five and forty-five degrees. Verified: zero, minus one over root two, and one over root two. The identity gives zero plus a half plus a half. Worth showing because it is the first item where a cosine comes out negative, and the negative is not an error.
• Exercise 11.1 Q2 (Part II p. 381). A line making equal angles with all three axes. Verified: if all three cosines are equal, three times the common square is one, so each cosine is plus or minus one over root three, and the whole triple takes one sign or the other together. The common answer a student writes is a third, from dividing one by three instead of taking a square root, and the sign choice is usually dropped as well. Both errors are worth naming.
• Exercise 11.1 Q5 (Part II p. 381). The three sides of the triangle with vertices three, five, minus four; minus one, one, two; and minus five, minus five, minus two. Verified: the first side's gaps are minus four, minus four, six, of length root sixty-eight; the second side's are minus four, minus six, minus four, also of length root sixty-eight; the third side's are eight, ten, minus two, of length root one hundred sixty-eight. Divide each triple by its own length. Note in passing that two of the three sides come out the same length — the triangle is isosceles — which the question does not mention and which makes a nice thirty-second aside.
• The chapter frontispiece (Part II p. 377). The opening page carries a QR code marked with the Part II catalogue number and the chapter number, an epigraph about invention owing more to imagination than to reasoning attributed to A. De Morgan, and a portrait captioned Leonhard Euler with the dates 1707 to 1783. The epigraph and the portrait are two different people — caption them separately or the explanation will appear to attribute the line to Euler. There is a footnote pointing at an NCERT laboratory handbook from 2005.
• The opening announcement (§11.1, Part II p. 377). It lists what the chapter will cover, and four of the listed items are about planes: the equations of planes, the angle between two planes, the angle between a line and a plane, and how far a point lies from a plane. Verified as absent: see the rationalisation note below.

Figures to have open

• The axis frame with one directed line out of the origin and three angle arcs. This is a redraw of Fig 11.1 (Part II p. 378) with the dashed projection box kept and the label r dropped, since §11.2 does not define it. It carries sections 1, 2 and 3 and should be the same drawing each time, with only the arrowhead reversing in section 3.
• A two-panel redraw of Fig 11.2 (Part II p. 379) for section 8: panel (a) as the chapter draws it, with the origin labelled — the printed panel (a) does not label it — and panel (b) enlarged, carrying both gamma marks, the short vertical stub at the lower-left vertex, and a right-angle square at the foot vertex, which the printed figure omits.
• Three small axis-frame thumbnails for section 9, one per axis, sized identically so the three triples read as peers. Not in the book.
• A slide for section 6 showing the identity as it arrives on the page — a fragment of derivation, the word before it, and no supporting line. This is a redraw, not a page scan.
• No figure is needed for sections 4, 10, 11 or 12; those are text and arithmetic.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 11 "Three Dimensional Geometry", §11.1 Introduction, Part II p. 377
• §11.2 Direction Cosines and Direction Ratios of a Line, the recalled definition and the supplement sentence, Part II p. 377
• Fig 11.1, the two-sets note and the Remark, Part II p. 378; the identity as it is used mid-derivation, Part II p. 378
• §11.2.1 Direction cosines of a line passing through two points, with Fig 11.2 panels (a) and (b), Part II pp. 379–380
• Examples 1, 3 and 4, Part II pp. 380–381; Exercise 11.1 questions 1, 2 and 5, Part II p. 381
• Summary, the first three bullets, Part II p. 391