Lines and planes in space
Could two straight lines cross twice? When two flat surfaces cut through each other, what shape is the place where they meet? And out in space, must two lines that never meet be parallel?
Where you are 8 in Points, lines and planes
Why this exists
A triangle, a circle and two crossing lines all fit on one flat sheet. A box, a ball and the room you are sitting in do not, and no amount of flat geometry describes them. Geometry needs a setting that holds many planes at once, stacked, tilted and crossing, with points above and below each: space. It also needs words that keep the skin of an object apart from the object, since the paint on a box and the sand inside it are measured in different ways.
Every figure in space is built from lines and planes, and the first thing to know about any two of them is where they touch. If two lines could cross in two places, or two walls could meet in a scattered handful of points, you could never say how many corners a shape has, or whether an edge you see is one straight edge. The answer is short, and it is what lets you name the corners and edges of a box, find where two graphs cross, and later describe every solid there is.
On a flat sheet of paper, two lines either cross or run side by side for ever, and the second kind are parallel. On paper, “never meet” and “parallel” mean the same thing. In space they do not. A road running east over a bridge and a road running north underneath it never meet, since one is always above the other, but nobody would call two roads at right angles parallel. Space has a third way for two lines to sit, and it needs its own name.
Definition
Space and figures in space
Space is the set of all points. It contains every plane, and points off every one of them.
A figure is any set of points. A plane figure is a figure whose points all lie in one plane: a triangle, a circle, two crossing lines. A space figure, or three-dimensional figure, is a figure whose points do not all lie in one plane.
A surface is a figure that is thin in one direction, like a plane or the skin of a ball. A closed surface has no edge and fits inside some large enough ball, like a sphere or the surface of a cube. A solid is a closed surface together with every point it encloses; the surface is its boundary. So a sphere is only the surface, every point at one distance from the center, and the sphere with everything inside it is a ball.
Figures also differ in dimension, the number of independent directions you can move in while staying inside the figure.
A point has dimension 0, a segment or line 1, a surface 2, and a solid, like space itself, 3: left and right, forward and back, up and down. Dimension 1 is measured by length, 2 by area and 3 by volume.
Intersections and parallels
The intersection of two figures is the set of all points that lie in both of them. The intersection of line \ell and line m is written \ell \cap m and read “\ell intersected with m”. Two figures intersect, or meet, when their intersection has at least one point. When it has none, the intersection is the empty set, written \varnothing.
A plane is named by one letter, such as plane P, or by three of its points not on one line, such as plane ABC; the line through A and B is \overleftrightarrow{AB}. Both go on forever, and a drawing shows only a piece of each.
Two lines that lie in one plane and never meet are parallel. A line and a plane that never meet are parallel, and so are two planes that never meet.
Skew lines
Two lines are skew when there is no plane that contains them both.
Parallel lines lie in one plane and do not meet, and lines that cross lie in one plane too (collinear and coplanar points). So skew lines are neither: they have no point in common, and no plane holds both (Figure 2).
Every box has skew lines. In Figure 3, \overleftrightarrow{AB} runs along the front bottom edge and \overleftrightarrow{CG} up the back right edge. They never meet and point in different directions: they are skew.
There is no symbol for “is skew to”. Skewness is about lines: two edges of a solid are called skew when the lines through them are.
Intuition
Where lines and planes touch
Lay two pencils on a table and cross them. However you slide them, they touch in one place, or, lined up side by side, nowhere. They cannot cross twice unless one bends. That is Figure 4.
Where a wall meets the floor, the two flat surfaces do not touch at a few scattered spots; they touch all along one straight crease. That is Figure 5.
Push a knitting needle through a sheet of card and it goes in at one hole. To touch the card at two points it must lie flat along it, and then it touches all the way. That is Figure 6.
What cuts space in two
A point cuts a line into two pieces: you cannot walk along the line from one to the other without stepping on it. A line cuts a plane in two, but a point cannot, since you walk around it. A plane cuts space in two, but a line cannot, since you step over it. A cutter two dimensions lower than what it cuts can always be walked around; the thinnest cutter that works is one dimension lower.
A closed surface does the same job, curved: the skin of a ball splits space into inside and outside, which is what it means for it to enclose a solid.
Why skew lines need space
Try to hold two lines with a flat sheet of card. For crossing lines, or parallel ones, you can. For the lines of Figure 3 you cannot: swing the card about \overleftrightarrow{AB} like a door on its hinge, and at each position it cuts \overleftrightarrow{CG} at one point or misses it, but never lies along it.
A drawing cannot show this by itself. The page is flat, so lines drawn in different directions cross somewhere on it. In Figure 3 that crossing is two points of space, one on each line, with the point of \overleftrightarrow{CG} behind the point of \overleftrightarrow{AB}. The break in \overleftrightarrow{CG} is how the drawing says so.
Results
The space postulate and space separation
Space postulate. Space contains at least four points that do not all lie in one plane.
It is what makes space more than a plane. The ways points, lines and planes meet, set out next, and the facts about collinear and coplanar points all hold in it.
Space separation. A plane \mathcal{F} divides the points of space that are not on it into two half-spaces, one on each side. Two points A and B not on \mathcal{F} are on the same side exactly when \overline{AB} does not meet \mathcal{F}, and on opposite sides exactly when it does.
Measure follows dimension. The frame, the surface and the inside of one object are three figures, of dimension 1, 2 and 3, measured in \text{cm}, \text{cm}^2 and \text{cm}^3. None can stand in for another.
How two lines, a line and a plane, and two planes meet
Two lines. Two different lines meet in at most one point. If the two lines lie in one plane, they either meet in exactly one point or they are parallel:
A line and a plane. If two different points of a line lie in a plane, then the whole line lies in the plane. So a line and a plane meet in exactly one of three ways: the line lies in the plane, the line does not lie in the plane and meets it in exactly one point, or they are parallel and do not meet.
Two planes. If two different planes meet, their intersection is exactly one line. Otherwise the planes are parallel:
Missing from every list: two points, three points, any scattered set.
Skew lines
Three possibilities. Two different lines in space do exactly one of these: they intersect in one point, they are parallel, or they are skew.
The piercing test. Suppose a line m meets a plane P at exactly one point X, and a line \ell lies in P but does not pass through X. Then \ell and m are skew.
Lines in parallel planes. If one line lies in a plane and a second line lies in a parallel plane, the two lines are either parallel or skew. They cannot intersect.
Parallel to the same line. If two different lines are each parallel to a third line, they are parallel to each other. In one plane this follows from the parallel postulate; it stays true in space, where the three lines need not share a plane, but that proof is beyond this lesson, so the fact is used here without proof. In a box, for instance, \overleftrightarrow{AB} and \overleftrightarrow{HG} are parallel because both are parallel to \overleftrightarrow{DC}.
The angle between skew lines. Skew lines make no angle at a point, so one of them is moved: take a line through a point of \ell parallel to m. The angle it makes with \ell (the one that is 90^\circ or less) is the angle between \ell and m. That it does not depend on the point chosen is another fact used here without proof.
Why it is true
Space separation
The space postulate is assumed, not proved: it says you are working in three dimensions, not two. Space separation is usually a postulate too, but you can see why it holds. Give every point a height above \mathcal{F}, negative below it. Along \overline{AB} the height changes at a steady rate, so the point a fraction t of the way from A to B has height
If h_A and h_B have the same sign, every height between them has that sign too, so the segment never reaches height 0 and misses the plane. If the signs differ, the height passes steadily from one to the other and is 0 somewhere, on \mathcal{F}. So “same sign” and “segment misses the plane” are one condition, and there are exactly two sides.
Lines and planes meet at most in a point or a line
Each result leans on the starting facts of geometry: through two different points there is exactly one line; through three points not on one line there is exactly one plane (see Collinear and coplanar points); and a plane is flat, meaning that if two different points of a line lie in a plane, the whole line does. That last fact is a postulate, accepted rather than proved.
Two different lines meet at most once. Suppose different lines \ell and m shared two different points A and B. Then each is a line through A and B. There is only one such line, so \ell = m, which contradicts their being different.
A line not in a plane meets it at most once. If line \ell shared two different points with plane P, flatness would put the whole of \ell inside P. So a line that does not lie in P shares at most one point with it.
Two different planes that meet share exactly a line. Suppose different planes P and Q share two different points A and B. By flatness all of \overleftrightarrow{AB} lies in P, and for the same reason in Q, so the whole line is shared.
Nothing off that line is shared. If a point C not on \overleftrightarrow{AB} were in both planes, then A, B and C, three points not on one line, would lie in two different planes. Only one plane passes through three such points, so that cannot happen, and P \cap Q = \overleftrightarrow{AB} exactly.
What the argument cannot supply is the second shared point: that two different planes sharing one point must share another. School geometry takes that as a postulate too.
The facts about skew lines
Three possibilities. If two different lines share a point, they intersect, and in one point only, as shown just above. If they share none, either some plane holds both, and they are parallel, or none does, and they are skew. No pair is two of these at once: intersecting lines meet and the others do not, and parallel lines share a plane while skew lines do not.
The piercing test. Suppose, to the contrary, that some plane Q contains both \ell and m.
The point X is on m, so X is in Q. So Q contains the line \ell and the point X, which is not on \ell. Now use this fact: through a line and a point not on that line there is exactly one plane. To see it, take two points A and B of \ell. The only line through A and B is \ell, and X is off it, so A, B, X are noncollinear, and exactly one plane contains three noncollinear points. Any plane containing \ell and X contains A, B and X, so it is that plane. Both P and Q contain \ell and X, so Q is P.
That means m lies in P, and then m meets P at every one of its points, not at X alone. This contradicts what we were told, so no such plane Q exists, and \ell and m are skew.
Lines in parallel planes. Parallel planes share no point, so a line in one and a line in the other share no point either. They cannot intersect, which leaves parallel or skew.
Worked examples
Plane figure or space figure
Say whether each is a plane figure or a space figure.
(a) Two lines that cross.
(b) The four corners of a sheet of paper lying on a desk.
(c) The eight corners of a box.
Solution(a) Two crossing lines lie in the plane through the crossing point and one more point on each line. Plane figure.
(b) The corners lie in the plane of the desk. Plane figure: four points are a space figure only when no plane holds them all.
(c) The only plane through the four bottom corners is the plane of the bottom face, and the top corners lie above it. Space figure.
The test is always whether one plane holds every point, however the figure is drawn.
One box, three figures
A box is 5 cm wide, 3 cm tall and 4 cm deep. Find the total length of its edges, the area of its surface and the volume of the solid box.
The 12 edges come in fours, one four for each length. They have dimension 1, so a length:
The surface is six rectangles in pairs: top and bottom 5 \times 4, front and back 5 \times 3, the sides 4 \times 3. It has dimension 2, so an area:
The solid has dimension 3, so a volume:
Three figures, three units: wire for the frame, paper to wrap it, sand to fill it.
Naming intersections on a box
Use the box in Figure 9. Name each intersection: (a) plane ABCD and plane BCGF; (b) \overleftrightarrow{AE} and plane EFGH; (c) plane ABFE and plane DCGH; (d) the three planes ABCD, ABFE and ADHE.
(a) The bottom face and the right face share the points B and C. Two different planes that meet share exactly the line through any two of their common points, so the intersection is \overleftrightarrow{BC}: the whole line, not just the edge you can see.
(b) The line runs up the front left edge. It does not lie in the top plane, since A is below it, so it meets that plane at most once. It meets it at E, so the intersection is the point E.
(c) The front and back faces stay the same distance apart however far they are extended, so their planes never meet and the intersection is \varnothing.
(d) Planes ABCD and ABFE share \overleftrightarrow{AB}. Plane ADHE contains A but not B, so that line does not lie in it and meets it only at A. The three planes share the single point A.
Where two lines in the coordinate plane meet
Find every point shared by the lines y = 2x + 1 and y = -x + 7.
SolutionThe slopes differ, so these are two different lines and share at most one point. A shared point has the same y on both lines, so set the two expressions equal:
Then y = 2(2) + 1 = 5. Check on the other line: -2 + 7 = 5. The lines meet at (2, 5) and nowhere else.
Two shared points mean one line
Line \ell passes through (1, 3) and (4, 9). Line m passes through (1, 3) and (7, 15). Do these lines cross at one point, or are they the same line?
SolutionThe slope of \ell is \dfrac{9 - 3}{4 - 1} = \dfrac{6}{3} = 2, and it passes through (1, 3), so its equation is y = 2x + 1.
Test m’s other point on \ell: 2(7) + 1 = 15. So (7, 15) lies on \ell as well.
Now \ell and m share the two different points (1, 3) and (7, 15). Only one line passes through two different points, so \ell and m are one line, y = 2x + 1, and share every point.
Counting crossings
Five lines lie in one plane. No two are parallel and no three pass through one point, as in Figure 10. How many points of intersection are there?
Every crossing point belongs to a pair of lines. The lines lie in one plane and no two are parallel, so every pair meets, and two different lines cannot meet twice, so every pair meets exactly once. No three lines pass through one point, so no point belongs to two different pairs. The number of points is therefore the number of pairs.
Each of the 5 lines pairs with the 4 others, giving 5 \times 4 = 20, but that counts every pair twice:
There are 10 points, the ten dots in Figure 10. The same argument gives \dfrac{n(n - 1)}{2} points for n such lines.
How many planes five points fix
Five points are placed in space so that no four of them lie in one plane. How many planes pass through three of them?
No three of the points lie on one line: that line and any fourth point would lie in one plane, which is ruled out. So every three of the points lie in exactly one plane. And two different sets of three never give the same plane, since between them they hold at least four points. So count the ways to choose 3 of 5 without regard to order:
The hypothesis does the work. Four noncoplanar points, as in Figure 11, give \binom{4}{3} = 4 planes, the faces of a triangular pyramid; five points in one plane give just one.
How three planes can meet
Three different planes are given. What can the set of points they all share look like?
SolutionStart with two of them. If they are parallel, no point lies in all three. Otherwise they meet in a line \ell, and the points shared by all three are where the third plane meets \ell. If the third plane contains \ell, all three share the whole line, like pages of an open book along the spine (Figure 12, top). If \ell does not lie in it but meets it, they share that one point, like a floor and two walls at a corner (Figure 12, bottom). If \ell is parallel to it, they share nothing.
So three different planes share a line, a single point, or nothing, never exactly two points.
Four planes and a pyramid
Four different planes are given. Every two of them meet, and no three contain a common line. How many different lines of intersection are there?
SolutionEach pair is two different planes that meet, so it meets in exactly one line. With 4 planes there are \dfrac{4 \times 3}{2} = 6 pairs.
Could two pairs give the same line? Two different pairs involve at least three different planes, so that line would lie in three of them, which is ruled out. So there are 6 different lines.
A triangular pyramid shows this: its 4 faces lie in 4 such planes, and each of its 6 edges lies on the line where two faces meet.
Which side of the floor
The floor of a building is a plane \mathcal{F}. Point A is 6 m above the floor and point B is 2 m below it, in the basement. Does \overline{AB} meet the floor? If \overline{AB} is 12 m long, how far from A is the meeting point?
Heights 6 and -2 have opposite signs, so by space separation \overline{AB} meets the floor, at a point X. Along the segment the height falls steadily from 6 to -2, a drop of 8, and reaches 0 after a drop of 6, which is
of the way from A to B. On a segment 12 m long, that is \displaystyle \frac{3}{4} \times 12 = 9 m from A, and so 3 m from B.
Check. The heights and the pieces are in the same ratio: 6 : 2 and 9 : 3 are both 3 : 1.
Cutting space with planes
One plane cuts space into 2 parts. (a) Find every number of parts that three different planes can cut space into. (b) Four planes are placed so that every three of them meet in exactly one point, and no point lies on all four. Into how many parts do they cut space?
(a) If all three planes are parallel they make 4 layers. If exactly two are parallel, those make 3 layers and the third cuts each in two: 6.
If no two are parallel, each pair meets in a line. Any two of these lines share a plane, so they meet or are parallel, and a point where two meet lies on all three planes. So the three lines are one line, and the planes are pages of an open book making 6 wedges; or three different parallel lines, and the planes form an endless triangular tube with 1 part inside, 3 across its faces and 3 beyond its edges, 7 in all; or three lines through one point, like the floor and two walls of a room extended, making 2 \times 2 \times 2 = 8.
So three planes make 4, 6, 7 or 8 parts, and never 5.
(b) The first three planes meet in one point, making 8 parts. The fourth meets each of them in a line. No two of these three lines are parallel or the same, or three planes would not meet in exactly one point, and they do not share a point, or it would lie on all four planes. Such lines cut the fourth plane into 7 pieces: two crossing lines make 4, and the third, cut into 3 by the other two, adds 3. Each flat piece splits a different old part in two, so:
This placement gives the most parts possible. The most parts n cutters can make climbs one dimension at a time:
For n = 4 planes that is 1 + 4 + 6 + 4 = 15, as found.
Sorting pairs of lines in a box
Use the box in Figure 3. Decide whether each pair of lines intersects, is parallel, or is skew. (a) \overleftrightarrow{AB} and \overleftrightarrow{EF}. (b) \overleftrightarrow{AB} and \overleftrightarrow{BF}. (c) \overleftrightarrow{AB} and \overleftrightarrow{FG}. (d) \overleftrightarrow{DH} and \overleftrightarrow{EF}.
Solution(a) \overline{AB} and \overline{EF} are opposite sides of the rectangle ABFE: one plane, no meeting point. The lines are parallel.
(b) The two edges share the corner B, so the lines intersect at B.
(c) \overleftrightarrow{AB} lies in the floor and \overleftrightarrow{FG} in the top, which are parallel planes, so the lines are parallel or skew. \overleftrightarrow{AB} runs left to right and \overleftrightarrow{FG} front to back, so they are not parallel: the lines are skew.
(d) The upright line \overleftrightarrow{DH} meets the plane of the top face EFGH only at the corner H. The line \overleftrightarrow{EF} lies in that plane, and H is not on it. By the piercing test, the lines are skew.
How many edges are skew to one edge
Find the edges of the box in Figure 3 that are skew to \overline{AB}. Then count the pairs of skew edges in the whole box.
SolutionFour of the other 11 edges share an endpoint with \overline{AB} and meet it: \overline{AD}, \overline{AE}, \overline{BC} and \overline{BF}. Three are parallel to it: \overline{DC} and \overline{EF}, opposite sides of a face, and \overline{HG}, which is parallel to \overline{DC} and so to \overline{AB}.
That leaves four: \overline{EH}, \overline{FG}, \overline{CG} and \overline{DH}. Example 12 showed \overline{FG} is skew to \overline{AB}, and the same two tests settle the other three.
Every edge of a box sits the way \overline{AB} does, with four skew edges. With 12 edges, the product 12 \times 4 counts each skew pair twice, once from each of its edges. So the box has
pairs of skew edges.
A line through the inside of the box
In Figure 3, the line \overleftrightarrow{AG} runs from the front bottom left corner to the back top right corner, through the inside of the box. Which edges are skew to it?
SolutionThe six edges ending at A or G meet \overleftrightarrow{AG}. For the other six the piercing test works with the point A, since \overleftrightarrow{AG} meets the plane of each of the three faces at A in that one point only. \overline{BC} and \overline{CD} lie in the floor, \overline{BF} and \overline{EF} in the front face, \overline{DH} and \overline{EH} in the left face ADHE, and none of them passes through A.
So all six, \overline{BC}, \overline{CD}, \overline{BF}, \overline{DH}, \overline{EF} and \overline{EH}, are skew to \overleftrightarrow{AG}.
How far apart two skew lines are
The box in Figure 3 has AB = 6, BC = 4 and AE = 3. Find the shortest distance between \overleftrightarrow{AB} and \overleftrightarrow{CG}, and between \overleftrightarrow{AB} and \overleftrightarrow{EH}.
Solution\overleftrightarrow{AB} lies in the plane of the front face and \overleftrightarrow{CG} in the plane of the back face. Those planes are parallel and 4 apart, since every edge running from front to back has length BC = 4. Any segment from a point of one line to a point of the other joins the two planes, so it is at least 4 long. The edge \overline{BC} joins B on the first line to C on the second and is exactly 4 long. So the shortest distance is 4.
For the second pair, \overleftrightarrow{AB} lies in the floor and \overleftrightarrow{EH} in the top, parallel planes 3 apart. The edge \overline{AE} joins A on the first line to E on the second and is 3 long. So the shortest distance is 3.
Both shortest segments, \overline{BC} and \overline{AE}, are perpendicular to both lines. In fact any two skew lines have exactly one segment perpendicular to both, and it is the shortest between them. That general fact is stated here without proof; the argument above covers only lines in two parallel faces of a box.
The angle between two skew face diagonals
In the cube of Figure 15, find the angle between \overleftrightarrow{AF} and \overleftrightarrow{BG}.
The lines are skew: \overleftrightarrow{BG} meets the plane of the front face only at B, and \overleftrightarrow{AF} lies in that face and misses B. So look for a line through A parallel to \overleftrightarrow{BG}. The edges \overline{AB} and \overline{HG} are parallel (Example 13) and of equal length, so ABGH is a parallelogram, and its opposite sides \overline{AH} and \overline{BG} are parallel. So the angle between the skew lines is the angle between \overleftrightarrow{AF} and \overleftrightarrow{AH}, which is \angle FAH.
Each side of triangle AFH is a diagonal of one face of the cube (front, left, top). The faces are equal squares, so the diagonals are equal, triangle AFH is equilateral, and each of its angles is 60^\circ.
The angle between \overleftrightarrow{AF} and \overleftrightarrow{BG} is 60^\circ.
Opposite edges of a tetrahedron
A tetrahedron is a solid with four triangular faces, four corners A, B, C, D, and six edges. Show that the opposite edges \overline{AB} and \overline{CD} are skew, and find how many pairs of skew edges a tetrahedron has.
SolutionA plane holding both \overleftrightarrow{AB} and \overleftrightarrow{CD} would hold all four corners, and the corners of a tetrahedron are not coplanar, or the solid would be flat. So the lines are skew.
Six edges make \displaystyle \frac{6 \times 5}{2} = 15 pairs. Each edge shares a corner with 4 others, so \displaystyle \frac{6 \times 4}{2} = 12 pairs intersect. The rest are the opposite pairs, just shown skew:
The 3 skew pairs are \overline{AB} with \overline{CD}, \overline{AC} with \overline{BD}, and \overline{AD} with \overline{BC}. A tetrahedron has no parallel edges at all.
Boundaries
The figures must be different. A line shares every one of its points with itself, and so does a plane. “At most one point” is about two different lines.
Lines and planes, not segments and faces. Two segments can miss each other though their lines cross, and two faces of a box can fail to touch though their planes meet. Extend to the full line or plane before using these results. The same goes for skewness: two edges that miss each other are not skew if their lines meet further out.
“One point or parallel” needs one plane. Two different lines meet at most once in space too, but two lines in space that do not meet need not be parallel: they can be skew. In one plane, two lines either meet or are parallel, so skew lines need space; and in space, “never meet” is not enough for parallel, since parallel lines must also share a plane.
Straightness is the whole reason. On a globe, where great circles play the part of lines, any two different great circles cross twice, at opposite points. The first result holds only because two different points fix one line.
A space figure is defined by its points, not its picture. A triangle drawn tilted is still a plane figure, and two edges that cross in a drawing may never meet in space. Where two pencil lines cross on the page says nothing about space; whether lines meet, or are skew, has to come from the solid itself.
A cutter too thin, or with an edge, separates nothing. A line or a point cannot split space, and nor can a disc or a plane with a hole in it: you go around the edge or through the hole.
A surface must be closed to enclose a solid. A bowl or a sheet encloses nothing. A plane has no edge and does split space, but both halves run on forever, so it encloses no solid.
The angle between skew lines is borrowed. It is taken from a parallel copy of one line, and it is more than 0^\circ (lines at 0^\circ would be parallel) and at most 90^\circ.
Three is not the only dimension. The patterns here (the cutter one dimension lower, one more binomial term per dimension) carry on in spaces of any dimension.
Where this goes next
Solving a pair of linear equations finds the point where two lines meet: the solution is the crossing point of the two graphs (Solving systems by graphing), and the parallel and same-line cases are the systems with no solution or infinitely many (Parallel and identical lines). One dimension up, an equation in three variables draws a plane, and the ways three planes meet are exactly the ways a system in three variables can have one solution, infinitely many, or none.
A point in space needs three coordinates, which is where coordinates and distance in three dimensions begin. With coordinates, the tests for skew lines become calculations. Distinct lines whose direction vectors point the same way are parallel; for the rest you solve for a common point, and if there is none they are skew. The cross product gives the direction of the segment perpendicular to both, the one that measured the distance in Example 15.
Solid figures are described by these same meetings: faces meet in edges, and edges meet at corners. The solids themselves get names and parts: three-dimensional shapes, faces, edges and vertices, prisms and pyramids and polyhedra, whose edges fall into the same three kinds, intersecting, parallel and skew. The angle pairs made by a transversal all live in one plane; a line crossing two skew lines never lies in one plane with both, and none of those results apply. A cross section is where one plane meets each face of a solid, a plane figure cut from a space figure. And the split between a surface and a solid is exactly the split between surface area and volume.
Exercises
Plane figure or space figure
Say whether each is a plane figure or a space figure, and why.
- A triangle.
- Three points that do not lie on one line.
- Four points, three on the floor of a room (not on one line) and the fourth on the ceiling.
- The surface of a cube.
- A circle together with a point that is not in the circle’s plane.
Dimension, surface and solid
- Give the dimension of each: (a) one edge of a table top, (b) the top surface of the table, (c) a brick, (d) the corner where two walls and the floor meet.
- A cube has edges of 6 cm. Find the total length of its edges, the area of its surface and its volume.
- A box is 10 cm by 6 cm by 2 cm. Find the total length of its edges, the area of its surface and its volume.
- A solid cube with edges of 4 cm is painted on the outside, then cut into cubes with 1 cm edges. How many small cubes have no paint on them, and how many have some?
Reading a box
Use the box in Figure 9 for Exercises 10 to 14.
- Name the intersection of plane ABFE and plane BCGF.
- Name the intersection of \overleftrightarrow{CG} and plane ABCD.
- Name the intersection of plane ADHE and plane BCGF.
- Name three planes, each containing a face of the box, that meet at point G.
- Name two planes, each containing a face of the box, whose intersection is \overleftrightarrow{DH}.
Lines in the coordinate plane
Find every point the two lines share.
- y = x + 4 and y = 3x - 2
- y = -x + 5 and \displaystyle y = \frac{1}{2}x - 1
- y = 2x - 1 and y = 2x + 5
- y = -2x + 9 and 2y = -4x + 18
- The line through (0, 1) and (2, 7), and the line through (1, 4) and (3, 10).
Counting intersections and planes
- Four lines lie in one plane, no two parallel and no three through one point. How many points of intersection are there?
- Seven lines lie in one plane, no two parallel and no three through one point. How many points of intersection are there?
- Some lines lie in one plane, no two parallel and no three through one point, and they make 45 points of intersection. How many lines are there?
- Six lines lie in one plane. No three pass through one point, and exactly one pair of them is parallel. How many points of intersection are there?
- Five different planes are given. Every two of them meet, and no three contain a common line. How many different lines of intersection are there?
- Seven points are placed so that no four lie in one plane. How many planes pass through three of them, and how many lines through two?
Which side of a plane
- Point A is 5 m above the floor, B is 3 m above it and C is 4 m below it. Which of \overline{AB}, \overline{AC} and \overline{BC} meet the floor? For \overline{AC}, what fraction of the way from A is the meeting point?
- Point A is 9 m above the floor and B is 3 m below it, and \overline{AB} is 20 m long. How far from A does \overline{AB} meet the floor?
Cutting space
- Into how many parts do five parallel planes cut space?
- The floor, the ceiling and one wall of a room are each extended into a whole plane. Into how many parts do they cut space?
- Four different planes all pass through one line. Into how many parts do they cut space?
- What is the most parts that five planes can cut space into?
Always, sometimes or never
Decide whether each statement is always, sometimes or never true, and explain.
- Two different lines meet in exactly two points.
- A line and a plane meet in exactly one point.
- Two different planes meet in exactly one point.
- Three different planes have exactly one point in common.
- Two lines that lie in one plane and do not meet are parallel.
Sorting pairs of lines
Use the box in Figure 3. Say whether each pair of lines intersects, is parallel, or is skew.
- \overleftrightarrow{EF} and \overleftrightarrow{HG}
- \overleftrightarrow{AD} and \overleftrightarrow{DH}
- \overleftrightarrow{BF} and \overleftrightarrow{EH}
- \overleftrightarrow{AE} and \overleftrightarrow{CD}
- Name every edge of the box that is skew to \overline{BF}.
- The line \overleftrightarrow{BH} runs through the inside of the box. Name every edge that is skew to it.
- Are \overleftrightarrow{AC} and \overleftrightarrow{EG}, the diagonals of the floor and the top, skew? Explain.
Counting skew edges
Use the triangular prism in Figure 16 for Exercises 44 to 46.
- Name every edge that is skew to \overline{CF}.
- Name every edge that is skew to \overline{AB}.
- How many pairs of skew edges does the triangular prism have?
- A square pyramid has a square base ABCD and a top corner V joined to each corner of the base. How many pairs of skew edges does it have?
- A prism has a regular pentagon for its base. How many of its edges are skew to one of the upright edges joining the two pentagons?
Distances and angles between skew lines
For Exercises 49 and 50, a box is lettered as in Figure 3, with AB = 8, BC = 5 and AE = 2.
- Find the shortest distance between \overleftrightarrow{AD} and \overleftrightarrow{EF}.
- Find the shortest distance between \overleftrightarrow{BF} and \overleftrightarrow{DC}.
For Exercises 51 to 53, use the cube in Figure 15.
- Find the angle between \overleftrightarrow{AF} and \overleftrightarrow{DH}.
- Find the angle between \overleftrightarrow{AF} and \overleftrightarrow{CH}.
- Find the angle between \overleftrightarrow{AC} and \overleftrightarrow{BG}.
Reasoning
- Line \ell does not lie in plane P. Explain why \ell meets P in at most one point.
- A carpenter tests whether a board is flat by laying a straightedge across it in many directions and looking for light underneath. Explain why a gap anywhere shows the surface is not a plane.
- Different planes P and Q both contain the different points A and B. Explain why P \cap Q = \overleftrightarrow{AB} exactly.
- Explain why three points always make a plane figure, but four points need not.
- A point splits a line into two parts and a line splits a plane into two parts. Explain why a line does not split space into two parts.
- Explain the difference between a sphere and a ball. Which one has a volume?
- Points A and B are on opposite sides of a plane, and so are B and C. Show that A and C are on the same side.
- Explain why two lines that lie in one plane can never be skew.
- True or false: in space, two lines that never meet are parallel. Explain.
- Find three lines through edges of the box in Figure 3 such that every two of them are skew, and explain why each pair is skew.
- A line \ell is drawn on the floor of a room. A second line m rises straight up from the floor at a point X. When are \ell and m skew, and when do they meet?
Answers
- Plane figure. A triangle’s three vertices fix one plane, and its sides lie in it.
- Plane figure. Three points not on one line lie in exactly one plane.
- Space figure. The only plane through the three floor points is the floor, and the ceiling point is not in it.
- Space figure. The top and bottom faces lie in different parallel planes.
- Space figure. The circle fixes its own plane, and the extra point is off it.
- (a) 1, (b) 2, (c) 3, (d) 0. A segment, part of a plane, a solid, a single point.
- Edges 72 cm, surface 216 \text{ cm}^2, volume 216 \text{ cm}^3. The edges are 12 \times 6, the surface is 6 squares of 36, the volume is 6^3. Same number, different units: not equal.
- Edges 72 cm, surface 184 \text{ cm}^2, volume 120 \text{ cm}^3. Edges 4(10 + 6 + 2), surface 2(60 + 20 + 12), volume 10 \times 6 \times 2.
- 8 with no paint and 56 with some. Of the 4^3 = 64 small cubes, only the inner 2 \times 2 \times 2 block misses the surface, and 64 - 8 = 56.
- \overleftrightarrow{BF}. The front and right faces share the edge from B to F, and two different planes that meet share exactly the line through two of their common points.
- The point C. The line runs up the back right edge and does not lie in the bottom plane, so it meets that plane at most once, and it does so at C.
- \varnothing. The left face and the right face are parallel planes and never meet.
- Planes BCGF, DCGH and EFGH: the right face, the back face and the top face.
- Planes ADHE and DCGH: the left face and the back face, which share the edge from D to H.
- (3, 7). Set x + 4 = 3x - 2, so 6 = 2x and x = 3; then y = 3 + 4 = 7. Check: 3(3) - 2 = 7.
- (4, 1). Set \displaystyle -x + 5 = \frac{1}{2}x - 1, so \displaystyle 6 = \frac{3}{2}x and x = 4; then y = -4 + 5 = 1. Check: \displaystyle \frac{1}{2}(4) - 1 = 1.
- No points. Setting 2x - 1 = 2x + 5 gives -1 = 5, which is never true. Same slope, different intercepts: the lines are parallel.
- Every point of the line. Dividing 2y = -4x + 18 by 2 gives y = -2x + 9, the same equation, so the two lines are one line.
- Every point of the line y = 3x + 1. The first line has slope \displaystyle \frac{7 - 1}{2 - 0} = 3 and y-intercept 1. Both (1, 4) and (3, 10) lie on it, since 3(1) + 1 = 4 and 3(3) + 1 = 10, and only one line passes through two different points.
- 6. Every pair meets once and no point is shared by two pairs: \displaystyle \frac{4 \times 3}{2} = 6.
- 21. \displaystyle \frac{7 \times 6}{2} = 21.
- 10 lines. Solve \displaystyle \frac{n(n - 1)}{2} = 45, so n^2 - n - 90 = 0 and (n - 10)(n + 9) = 0. A number of lines cannot be -9, so n = 10.
- 14. Six lines make \displaystyle \frac{6 \times 5}{2} = 15 pairs, and every pair meets once except the parallel pair, so 15 - 1 = 14.
- 10. There are \displaystyle \frac{5 \times 4}{2} = 10 pairs, each meeting in one line, and no two pairs give the same line, since that line would lie in three planes.
- 35 planes and 21 lines. \binom{7}{3} = 35 and \binom{7}{2} = 21. No three of the points are on a line, or those three and any fourth would share a plane, so no two pairs share a line and no two sets of three share a plane.
- \overline{AC} and \overline{BC} meet the floor; \overline{AB} does not. \overline{AC} meets it \displaystyle \frac{5}{9} of the way from A: only C has negative height, and along \overline{AC} the height drops 5 + 4 = 9, reaching 0 after 5.
- 15 m. The height drops 9 + 3 = 12 and reaches 0 after 9, which is \displaystyle \frac{9}{12} = \frac{3}{4} of the way, and \displaystyle \frac{3}{4} \times 20 = 15.
- 6 parts. Each new parallel plane splits one layer, so n parallel planes make n + 1 layers.
- 6 parts. The floor and ceiling make 3 layers, and the wall cuts each in two: 2 \times 3 = 6.
- 8 parts. Planes through one line make wedges around it like the pages of a book, and n of them make 2n: 2 \times 4 = 8.
- 26 parts. 1 + 5 + \binom{5}{2} + \binom{5}{3} = 1 + 5 + 10 + 10 = 26. Step by step: the fifth plane carries four lines cutting it into 1 + 4 + 6 = 11 pieces, and 15 + 11 = 26.
- Never. Two different lines share at most one point.
- Sometimes. It happens when the line does not lie in the plane and passes through it. The line could instead lie in the plane (infinitely many points) or be parallel to it (none).
- Never. Two different planes that meet share a whole line, which has infinitely many points.
- Sometimes. A floor and two walls share only a corner, but three pages of a book share a line, and three parallel planes share nothing.
- Always. That is the definition of parallel lines.
- Parallel. They are opposite sides of the rectangle EFGH.
- Intersecting, at D, the corner the two edges share.
- Skew. \overleftrightarrow{BF} meets the plane of the top face only at F, and \overleftrightarrow{EH} lies in that plane but does not pass through F.
- Skew. \overleftrightarrow{AE} meets the floor only at A, and \overleftrightarrow{CD} lies in the floor but does not pass through A.
- \overline{AD}, \overline{DC}, \overline{EH} and \overline{HG}. The edges at B and F meet it, and \overline{AE}, \overline{CG}, \overline{DH} are parallel to it.
- \overline{AD}, \overline{AE}, \overline{CD}, \overline{CG}, \overline{EF} and \overline{FG}. The six edges at B or H meet it; the piercing test handles the rest, as in Example 14.
- No. They are parallel. \overline{AE} and \overline{CG} are both parallel to \overline{BF}, so to each other, and equal in length, so ACGE is a parallelogram with opposite sides \overline{AC} and \overline{EG}.
- \overline{AB} and \overline{DE}. The edges at C and F meet \overline{CF}, and \overline{AD} and \overline{BE} are parallel to it.
- \overline{CF}, \overline{DF} and \overline{EF}. The four edges at A and B meet it, and \overline{DE} is parallel to it.
- 12. Each of the 6 triangle edges is skew to 3 edges (as in Exercise 45) and each of the 3 upright edges to 2 (as in Exercise 44), giving \displaystyle \frac{6 \times 3 + 3 \times 2}{2} = 12.
- 8. A base edge such as \overline{AB} is skew to the two slanted edges that miss A and B, namely \overline{VC} and \overline{VD}. A slanted edge such as \overline{VA} is skew to the two base edges that miss A. So the count is \displaystyle \frac{4 \times 2 + 4 \times 2}{2} = 8.
- 6. The upright edge meets the two pentagon edges at each of its ends (4 in all) and is parallel to the other 4 upright edges. The remaining 3 edges of the bottom pentagon and 3 of the top are skew to it.
- 2. \overleftrightarrow{AD} lies in the floor and \overleftrightarrow{EF} in the top, parallel planes 2 apart, and \overline{AE} joins them at right angles to both lines.
- 5. \overleftrightarrow{BF} lies in the front face and \overleftrightarrow{DC} in the back, parallel planes 5 apart, and \overline{BC} joins them.
- 45^\circ. \overleftrightarrow{DH} is parallel to \overleftrightarrow{AE}, so the angle is \angle FAE, and the diagonal of the square ABFE splits its right angle in half.
- 90^\circ. \overline{BC} and \overline{EH} are both parallel to \overline{AD} and have equal length, so BCHE is a parallelogram and \overleftrightarrow{CH} is parallel to \overleftrightarrow{BE}. The diagonals \overline{AF} and \overline{BE} of the square ABFE cross at its center, a point of \overleftrightarrow{AF}, at right angles.
- 60^\circ. \overleftrightarrow{BG} is parallel to \overleftrightarrow{AH} (Example 16), so the angle is \angle CAH. Triangle ACH has three face diagonals for sides, so it is equilateral.
- If \ell shared two different points with P, then, because a plane is flat, the whole of \ell would lie in P. It does not, so \ell shares at most one point with P.
- The straightedge rests on the surface at two different points, one near each end. If the surface were a plane, the whole line through those points would lie in it, so the edge would touch all the way along. A gap means a point of that line is off the surface, so it is not a plane.
- By flatness \overleftrightarrow{AB} lies in P and in Q. A shared point C off that line would put three points not on one line in two different planes, but only one plane passes through three such points. So the planes share the line and nothing else.
- Three points either lie on one line or fix exactly one plane, and either way a plane holds them all. A fourth point can be off that plane, and then no plane holds all four.
- Any point off the line can reach any other by going around the line. A line is two dimensions lower than space, too thin to separate it; the cutter has to be a surface.
- A sphere is the surface only, the points at one distance from the center; a ball is the sphere with every point inside it. The ball is a solid and has a volume; the sphere has only an area.
- Give heights above the plane. A and B have opposite signs, and so do B and C, so A and C both have the sign opposite to B’s. Then \overline{AC} never reaches height 0, and by space separation A and C are on the same side.
- Skew lines are defined as lines that no plane contains. If a plane contains both lines, they are by that fact not skew: they either meet or are parallel.
- False. Parallel lines never meet, but neither do skew lines, such as \overleftrightarrow{AB} and \overleftrightarrow{CG} in Figure 3. In space, parallel lines must also lie in one plane.
- For example \overleftrightarrow{AB}, \overleftrightarrow{CG} and \overleftrightarrow{EH}. \overleftrightarrow{CG} meets the floor only at C, which is off \overleftrightarrow{AB}, and meets the top only at G, which is off \overleftrightarrow{EH}: two piercing tests. \overleftrightarrow{AB} and \overleftrightarrow{EH} lie in the parallel floor and top and run in different directions, so they are skew as in Example 12(c).
- m meets the floor only at X. If \ell passes through X, the lines meet there. If \ell misses X, the piercing test says they are skew. They are never parallel.