Skew lines, what shortest distance can mean when two lines never meet, and computing it in both the skew and parallel cases

What to assume they know

• The equation of a line in space, in vector form and in symmetric Cartesian form — the second module
• The angle between two lines, and the condition that makes two lines parallel — the previous topic
• Coordinates of the corners of a rectangular box, from Class XI
• The idea of a shortest distance from a point to a line in the plane, and that it is measured along a perpendicular
• Deciding whether two lines meet by solving their equations simultaneously
• The perpendicular distance from a point to a line as the height of a triangle
• The cross product of two vectors, its length, and the fact that it is at right angles to both factors, from Chapter 10 of this volume
• The dot product, and the projection of one vector along a direction, from Chapter 10 of this volume
• A unit vector, and dividing a vector by its own length to make one
• Evaluating a three-by-three determinant by expanding along a row

What they should be able to do

• Sort two lines in space into the three cases the chapter names, and say which distance each case yields
• State what makes a pair of lines skew, in the chapter's own two-part form
• Explain why the skew case cannot occur between two lines in a plane
• Identify a skew pair on a concrete solid, and check both halves of the definition on it
• Say precisely what is being minimised when a shortest distance is sought
• Argue why the segment realising that minimum must be at right angles to both lines
• Read the chapter's skew-lines figure critically, knowing what a flat drawing cannot show
• Build the unit vector along the common perpendicular from the two direction vectors
• Follow the chapter's derivation, identifying the step where the unknown connecting segment cancels out
• State the vector formula for the skew case and read it as a projection
• Identify the printed error in the chapter's own displayed result for the skew case, and say how it should read
• Recognise the numerator as a scalar triple product, and read the Cartesian version as the same thing written as a determinant
• Say why the parallel case cannot use that formula, and derive the one it needs
• Work a skew pair end to end from two vector equations
• Work a parallel pair end to end, including the check that the two lines really are parallel
• Distinguish the three cases before starting any computation, and say which formula each case will need
• Interpret a computed distance of zero
• Locate the one place in the chapter where the angle between two skew lines is defined

Where it usually goes wrong

• "Two lines that do not meet are parallel." True in a plane, false in space, and this is the entire content of the section. A student carrying the plane intuition forward has no room in their head for the third case, and will try to apply the parallel formula to a skew pair.
• "Skew means slanted." It means neither parallel nor meeting. The word carries an everyday sense that has nothing to do with the definition, and students who guess from the word alone guess wrong.
• "Checking that two lines are not parallel is enough to call them skew." Both halves are needed. Two non-parallel lines in space may perfectly well meet, and then the distance is zero rather than something to compute.
• "The shortest distance is between the two nearest labelled points." It is between two points that generally carry no label at all and are not given in the question. The minimum is over every pair of points on the two infinite lines.
• "The joining segment only has to be square to one of the lines." Square to one gives the shortest join from a fixed point to that line, not the shortest join between the two lines. Both conditions are needed, and the sliding argument shows why.
• "The figure shows the four points lying in one plane, so they do." They cannot, and the drawing is a flattened projection. This is worth stating once, out loud, because every printed figure of a skew pair has the same problem and students quietly conclude the definition is inconsistent.
• "The two points in the formula have to be the feet of the common perpendicular." They can be any point on each line, which is exactly what the cancellation in the derivation buys. Believing otherwise leaves a student unable to start, because the feet are not given and finding them is harder than the original question.
• "The numerator is a length, so it cannot be negative." It is a number that can come out either sign, which is why the chapter wraps the whole quotient in modulus bars. Every one of the exercise items above has a negative numerator except two, and the sign is discarded at the end.
• "Parallel and skew use the same formula." They do not, for the good reason that a skew pair has one distinguished common perpendicular while a parallel pair has infinitely many, all of the same length — and mechanically, because the skew formula divides by zero when the two directions are parallel. That division is why §11.5.2 exists, and the chapter opens it without saying so.
• "The Cartesian formula is a second result to memorise." It is the vector formula written in components: the determinant is the triple product and the square root is the length of the cross product. One result, two dresses.
• "A distance of zero means I made a mistake." It means the two lines meet. The formula classifies and measures in one step, which the chapter does not point out.
• "Parallel lines are obvious from the equations." Only when the two direction vectors are written identically, as in Example 10. If one is a multiple of the other the pair is still parallel and the skew formula still fails, so the check has to be for proportionality, not for equality.
• "The order of the two lines matters." Swapping them reverses the sign of both the cross product and the anchor gap, so the numerator changes sign twice and the answer is unchanged. Worth ten seconds, because students who get a different sign from a friend assume one of them is wrong.
• "The printed formula on the page is the one to use." Not on Part II p. 387. Use the Summary's version, or the derivation on the facing page, both of which carry a difference where Part II p. 387 prints a cross.
• "The angle between skew lines is undefined." It is defined, by the same translate-to-a-common-point construction the previous topic used — but the chapter says so only in its Summary.

Questions to check understanding

• Classify a stated pair of lines as meeting, parallel or skew, and say which distance formula the case calls for
• Show that a named pair of edges or diagonals of a box is skew, checking both halves
• Explain why two lines drawn in one plane can never be skew
• State what quantity the shortest distance is the minimum of
• Argue that the segment realising the minimum is square to both lines
• Given a figure of a skew pair, say why the four drawn points cannot really be coplanar
• Say how the parallel case differs from the skew case in the number of perpendiculars available
• Locate a pair of skew lines on a solid other than the chapter's room
• Compute the shortest distance for a skew pair stated in vector form — the form of Example 9 and of Exercise 11.2 Q12 and Q14
• Compute it for two skew lines given in symmetric Cartesian form — the form of Exercise 11.2 Q13
• Extract anchors and directions from equations written with the parameter distributed, then compute — the form of Exercise 11.2 Q15
• Compute the distance between two parallel lines, stating the check that they are parallel — the form of Example 10
• Show that a stated pair of lines meets, by computing a distance of zero
• Explain why the choice of a point on each line does not affect the answer
• Rewrite the vector formula as a determinant and identify each row

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 three-way opening (§11.5, Part II p. 385). The section opens by disposing of two cases before naming the third. If the two lines meet, the answer is zero. If they run parallel, the answer is the length of a perpendicular dropped from a point of one onto the other. Only then does the chapter say that a third possibility exists. This ordering is the section's best pedagogical decision. — the skew case is defined by what it is not, and hearing what it is not first makes the definition land.
• The definition (§11.5, Part II p. 385). A pair that is neither meeting nor parallel is described as not lying in one plane, and named. Verified: the two descriptions agree — two distinct lines lie in a common plane exactly when they either meet or run parallel — but the chapter asserts the equivalence rather than arguing it. The explanation may run the argument in one sentence and should say it is supplying it.
• Fig 11.5, the room (Part II p. 385). Read off the printed page. The chapter sets up a rectangular room with edge lengths one, three and two along the three axes, and the figure draws the box in an axis frame with the origin at a floor corner. All eight corners carry letters — seven of them lettered in sequence, plus the origin — and not one carries coordinates; the three edge lengths appear only in the prose and nowhere in the drawing. Two lines are drawn across the box, each extended beyond it with an arrowhead at both ends, so the figure carries four arrowheads on the two lines plus three on the axes. See the note below on the corner count, which corrects an earlier reading of this same the printed page.
• The two lines named (Part II p. 386). One runs diagonally across the ceiling. The other starts at the ceiling corner sitting directly above one floor corner and runs diagonally down a wall. Verified from the figure and the stated edge lengths: taking the origin at the floor corner the axes meet, the ceiling diagonal joins the ceiling corner above the origin to the ceiling corner diagonally opposite it, so its direction is one, three, zero; the wall diagonal joins the ceiling corner above the far x-corner to the floor corner diagonally across that wall, so its direction is zero, three, minus two. The two directions are not proportional, so the lines are not parallel; and equating the two parametrised points forces both parameters to zero from the third coordinate and then the second, after which the first demands that zero equal one, so they never meet. Both halves check out, and the chapter states them without working either. Supply the working — it is the chapter's only concrete skew pair and it deserves the twenty seconds.
• What is being minimised (Part II p. 386). The chapter says the shortest distance is the smallest among the lengths of all segments joining one point of one line to one point of the other. Verified as a genuine minimum: both lines are infinite, so the set of candidate lengths is infinite; it is bounded below by zero and, for a skew pair, never reaches zero, and the minimum is attained. The chapter asserts the minimum exists and does not argue it.
• Why the minimising segment is square to both (Part II p. 386). The chapter states this in one sentence with no argument. Verified, and the argument is worth running: if the join were not at right angles to one of the lines, sliding its endpoint a little along that line would shorten it, because the foot of a perpendicular is nearer than any other point of a line. So a segment that cannot be shortened must be square to both. This is the one place in the section where a short argument replaces an assertion.
• Fig 11.6 (Part II p. 386). Read off the printed page. The lower line carries two named points and an arrowhead at each end; the upper line carries two named points and an arrowhead at each end. A segment joins the leftmost point of the lower line to the leftmost point of the upper one. A second segment runs vertically from the right-hand point of the lower line up to the right-hand point of the upper one, and carries an arrowhead partway along, so it is drawn as a directed vector. There is exactly one right-angle mark in the figure, at the foot on the lower line; the upper foot carries no mark at all. The chapter's own text calls the segment square to both directions, so the figure is one mark short of its own sentence. A redraw should mark both.
• The figure's unavoidable dishonesty (Fig 11.6, Part II p. 386). The four named points are drawn as a flat quadrilateral on the page. Verified as necessarily misleading: four points, two on each of two skew lines, cannot be coplanar — if they were, both lines would lie in that plane and the pair would not be skew. The drawing is a projection. This is the single most valuable thing a step-by-step treatment can add to this section.
• The setup for the derivation (§11.5.1, Part II p. 386, on the same Fig 11.6). Two lines are given in vector form. A point is taken on each — any point, the anchors of the two equations will do — and the connecting segment between the two feet of the common perpendicular is called by name. The chapter's whole argument turns on the observation that the length wanted is the projection of the arbitrary join onto the perpendicular direction. The figure has already been read in the two entries above.
• The unit vector (§11.5.1, Part II p. 386). Since the connecting segment is at right angles to both direction vectors, its direction is theirs crossed together, normalised to length one. Verified: the cross product is at right angles to both factors, and it is non-zero exactly because the two directions are not parallel — which is precisely what being skew guarantees. The chapter uses this fact in a subordinate clause and never states it as a result; Chapter 10 of this volume supplies it.
• The derivation (§11.5.1, Part II pp. 386–387). Five printed lines. The connecting segment is the distance times the unit vector. The cosine of the angle between the arbitrary join and the connecting segment is the dot product over the product of their lengths. Substituting and simplifying, the length of the arbitrary join cancels top and bottom, and so does the distance itself, leaving an expression that no longer mentions either. Verified, and this is the step the explanation exists to show: the arbitrary choice of one point on each line disappears from the answer, which is why any two anchors may be used. A student who does not see the cancellation believes the formula only works for the two special points at the feet of the perpendicular, and then does not know how to find them.
• The result, and a printed error in it (Part II p. 387). The chapter's displayed conclusion for the skew case prints the second bracket of its numerator as a cross product of the two anchor position vectors where their difference belongs. Read on the printed page and unambiguous — the glyph is the same multiplication cross used in the first bracket, three characters to its left. Verified as an error against three separate places in the chapter's own text: the derivation on the facing page, Part II p. 386, carries the difference at every step; Example 9 on Part II p. 388 computes the difference explicitly and substitutes it; and the Summary bullet on Part II p. 392 prints the difference. This is the single most damaging printed slip in the chapter, because it lands on the one line a student would copy into a formula sheet. Section 7 should show the printed line, mark the offending symbol and put the correct form beside it.
• The numerator as one number (Part II p. 387). Two vectors crossed, then dotted into a third. Verified: the result is a single number whose absolute value is the volume of the box spanned by the three vectors, and it vanishes exactly when the three lie in one plane — that is, exactly when the two lines meet. The chapter neither names this product nor draws the volume interpretation; both are not in the book and should be flagged. The explanation may still use the vanishing as a test, because it follows from the formula the chapter does print.
• The Cartesian version (Part II p. 387). Printed under a bold heading that carries no section number. A three-row determinant sits over a square root of three squared brackets, and the whole quotient sits inside outer modulus bars. Verified: the determinant's first row is the three coordinate gaps between the two anchors and the other two rows are the two direction-ratio triples, so it is the same triple product written differently; and the square root is the length of the cross product written out in components. Nothing new is happening. Say so — students treat this as a second formula to memorise.
• The parallel case as a different question (§11.5, Part II p. 385, and §11.5.2, Part II p. 387). Named in the opening sentence and given its own subsection later. Verified as a different question: for a parallel pair every perpendicular between the two lines has the same length, so any point of one will do as a starting point; for a skew pair there is exactly one segment realising the minimum and its endpoints are not free. The chapter states the parallel rule at the top of §11.5 and does not contrast the two situations. The contrast is added here and opens section 9.
• §11.5.2 and Fig 11.7 (Part II pp. 387–388). For a parallel pair, the two directions coincide, so the cross product of the two directions vanishes and the skew formula divides by zero. The chapter therefore starts again: take the join between the two anchors, take the angle it makes with the common direction, and read the perpendicular height off a sine. Verified: crossing the direction with the join gives a vector whose length is the product of the two lengths times that sine, and the perpendicular height is the join's length times the sine, so the height is the length of that cross product over the length of the direction. The chapter does not say why the skew formula fails here — it simply opens a new subsection. Section 9 supplies the reason and should say it is doing so.
• Fig 11.7 (Part II p. 387). Read off the printed page. Two parallel lines are drawn, each with a single arrowhead at its right-hand end. A point on the upper line and a point on the lower line are labelled with their position vectors in brackets beside them. A third point sits on the lower line, and a vertical segment joins it to the point on the upper line. Right-angle marks appear at both ends of that vertical segment, which is more than Fig 11.6 on the previous page manages. The angle between the join and the lower line is marked at the lower-left point. The direction vector itself is not drawn or labelled anywhere in the figure, though the surrounding text names it — a redraw should add it.
• Example 9 (Part II p. 388). Two lines in vector form, with anchors at the points one, one, zero and two, one, minus one, and directions two, minus one, one and three, minus five, two. Verified: the gap between anchors is one, zero, minus one; the cross product of the two directions is three, minus one, minus seven, whose length is the root of fifty-nine; the dot product of the two is three plus zero plus seven, which is ten; so the distance is ten over the root of fifty-nine. Also verified as genuinely skew: the directions are not proportional and the triple product is not zero, so the pair neither runs parallel nor meets — the chapter states neither check.
• Example 10 (Part II pp. 388–389). Two lines with the same direction two, three, six, anchored at one, two, minus four and three, three, minus five. Verified: the gap between anchors is two, one, minus one; crossing the direction into it gives minus nine, fourteen, minus four, whose length is the root of two hundred ninety-three; the direction's length is seven exactly; so the distance is the root of two hundred ninety-three over seven. The chapter's own solution opens with a parenthetical question mark asking the reader why the lines are parallel, and answers it nowhere. Answer it: the two direction vectors are identical character for character.
• The meeting case, and a test for it (Part II p. 385). The chapter disposes of it in half a sentence: the answer is zero. Verified: the shortest distance formula returns zero exactly when the two lines meet, so the same computation classifies the pair and measures it in one step. That is a genuinely useful thing to know before the arithmetic starts, and it is nowhere in the chapter.
• The five shortest-distance items, classified before any of them is computed (Exercise 11.2 Q12 to Q15 and Miscellaneous Exercise Q4, Part II pp. 390–391). Because two of the three cases need no formula at all. Verified by inspection of the direction triples: none of the five pairs is parallel, so all five need the skew formula, and none of the five turns out to meet.
• Exercise 11.2 Q12 (Part II p. 390). Anchors one, two, one and two, minus one, minus one; directions one, minus one, one and two, one, two. Verified: the gap is one, minus three, minus two; the cross product is minus three, zero, three, of length three root two; the dot product is minus three plus zero minus six, which is minus nine; the distance is nine over three root two, which is three over root two.
• Exercise 11.2 Q13 (Part II p. 390). Cartesian form; anchors minus one, minus one, minus one and three, five, seven; ratios seven, minus six, one and one, minus two, one. Verified: the gap is four, six, eight; the cross product is minus four, minus six, minus eight, of length two root twenty-nine; the dot product is minus sixteen minus thirty-six minus sixty-four, which is minus one hundred sixteen; the distance is one hundred sixteen over two root twenty-nine, which reduces to two root twenty-nine. Note the coincidence worth showing: the cross product comes out as exactly minus one times the gap, so the numerator is minus the squared length of the gap and the arithmetic collapses.
• Exercise 11.2 Q14 (Part II p. 390). Anchors one, two, three and four, five, six; directions one, minus three, two and two, three, one. Verified: the gap is three, three, three; the cross product is minus nine, three, nine, of length three root nineteen; the dot product is minus twenty-seven plus nine plus twenty-seven, which is nine; the distance is nine over three root nineteen, which is three over root nineteen.
• Exercise 11.2 Q15 (Part II p. 390). Both lines are written with their parameters distributed through the components rather than collected, so the anchors and directions have to be extracted first. Verified: the first line has anchor one, minus two, three and direction minus one, one, minus two; the second has anchor one, minus one, minus one and direction one, two, minus two. The gap is zero, one, minus four; the cross product is two, minus four, minus three, of length root twenty-nine; the dot product is zero minus four plus twelve, which is eight; the distance is eight over root twenty-nine. The extraction is the whole difficulty and it is the same skill as reading direction ratios off a rearranged Cartesian equation; give it its own beat.
• Miscellaneous Exercise Q4 (Part II p. 391). Anchors six, two, two and minus four, zero, minus one; directions one, minus two, two and three, minus two, minus two. Verified: the gap is minus ten, minus two, minus three; the cross product is eight, eight, four, of length twelve exactly; the dot product is minus eighty minus sixteen minus twelve, which is minus one hundred eight; the distance is one hundred eight over twelve, which is nine exactly. This is the only item in the chapter whose shortest distance is a whole number, and it should be the one the explanation finishes its arithmetic on.
• The Summary bullets (Part II pp. 391–393). Six belong here. Skew lines defined as neither parallel nor meeting, with a second sentence saying they lie in different planes; a bullet defining the angle between two skew lines; the shortest distance described as the segment square to both lines; the vector formula for the skew case, printed with the difference and so contradicting Part II p. 387; the Cartesian determinant; and the parallel formula. The angle bullet is not in the body — see the note below. Verified: the Summary's version of the Cartesian determinant carries no outer modulus bars, where the body's on Part II p. 387 does. A determinant can come out negative and a distance cannot, so the body's version is the safe one to show.

Figures to have open

• Three panels of two lines each for section 1 — meeting, parallel, and passing — drawn in one axis frame style and at one type size, since they are peers and sizing each to its own content would imply a hierarchy that is not there. Not in the book; the chapter illustrates none of the three cases at this point.
• A plane with two lines on it, and a third line lifting off it, for the second half of section 1. Not in the book, and the figure the chapter most needs and does not have.
• A redraw of Fig 11.5 (Part II p. 385) for section 2: the box in an axis frame with every lettered corner, both diagonals drawn and extended past the box with arrowheads at both ends. Add the three edge lengths as labels on the box — the printed figure carries none, and the prose gives them — and add coordinates to the two ceiling corners the diagonals start from, which the printed figure also lacks.
• A rotating redraw of Fig 11.6 (Part II p. 386), step by step, carrying sections 3 to 6 as one drawing that gains elements. It opens in section 3 unlabelled, with a spread of candidate joins between the two lines and the shortest picked out; section 4 names the four points and marks both right angles, where the printed figure marks only the one on the lower line; section 5 adds the two crossed direction vectors and the unit vector along the join; section 6 restores two or three alternative joins in a lighter weight so the cancellation has something to be about. This is the brief's most important figure and it is worth showing as a movement, because a still image cannot make the point section 4 exists to make. Drawing it once and adding to it is also what keeps the two halves of the explanation one video.
• A large two-line slide for section 7: the chapter's printed result above with the cross marked, and the corrected result below. Not in the book.
• A side-by-side for section 8 pairing the vector formula with the determinant, with each determinant row linked by an arrow to the vector it holds. Both panels must share one type size; sizing the determinant to its own content would make it read as the more important of two equal statements.
• A side-by-side for section 9: one skew pair with a single common perpendicular highlighted, one parallel pair with several equal perpendiculars drawn. Not in the book.
• A redraw of Fig 11.7 (Part II p. 387) for section 9, keeping both right-angle marks and the marked angle, and adding the direction vector as a labelled arrow — the printed figure names it in the text and does not draw it.
• A three-branch decision figure for section 11. Not in the book; the chapter sets the three cases out in prose only.
• No figure is needed for sections 10 or 12; those are arithmetic and a statement.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 11 "Three Dimensional Geometry", §11.5 Shortest Distance between Two Lines, the three cases and the definition of a skew pair, with Fig 11.5, Part II p. 385
• The named pair of lines on the room, the definition of the shortest distance and the sentence on the common perpendicular, Part II p. 386
• §11.5.1 Distance between two skew lines, the setup with Fig 11.6, the unit vector and the derivation, Part II p. 386
• The displayed result for the skew case and the unnumbered Cartesian heading with its determinant, Part II p. 387
• §11.5.2 Distance between parallel lines, with Fig 11.7, Part II pp. 387–388
• Examples 9 and 10, Part II pp. 388–389
• Exercise 11.2 questions 12 to 15, Part II p. 390; Miscellaneous Exercise question 4, Part II p. 391
• Summary, the skew-lines bullet, the angle-between-skew-lines bullet and the four shortest-distance bullets, Part II pp. 391–393