C. Geometry and Algebra of Vectors · Lines and PlanesThree Planes: Geometric ConfigurationsC3.3 C4.3About 70 minutesLearning goalsI can describe and sketch the eight configurations of three planes.I can organize configurations of lines and planes by whether and how they intersect.I can use the triple scalar product of the normals to decide whether three planes meet in a single point.I can identify the configuration of three given planes from their equations.Words to knowtriangular prism (configuration)Three planes that meet in pairs along three parallel, distinct lines, with no point common to all three.sheaf of planesThree or more planes that all contain one common line.redundant equationAn equation in a system that is a combination of the others, so it adds no new information.triple scalar productn₁ · (n₂ × n₃). For three normals, it is non-zero exactly when the normals are not coplanar.Floors, walls and tentsStand in the corner of a room: the floor and two walls meet in a single point. Look at three pages of an open book: they all share the spine, a line. Look at a pup tent: its two sloping sides meet along the ridge, each side meets the ground along a line, but no point is on all three surfaces at once.Three planes can be arranged in eight different ways. In this lesson you will picture all of them, sort them by how they intersect (a point, a line, a plane, or nothing) and use the normals to tell them apart. The next lesson, Solving Systems of Three Planes Algebraically, finds the intersections with algebra.one common pointz = 0x = 0y = 0n₁ · (n₂ × n₃) ≠ 0The floor and two walls of a room meet at a single corner. Here the planes are x = 0, y = 0 and z = 0, meeting only at the origin. Three planes meet in exactly one point when their normals are not coplanar.The eight configurationsStart with the normals n₁, n₂, n₃.If n₁ · (n₂ × n₃) ≠ 0, the three normals do not lie in one plane. Then the planes meet in exactly one point. This answers the curriculum question about the triple product: a non-zero value guarantees a single point of intersection.If n₁ · (n₂ × n₃) = 0, the normals are coplanar, and there is no single point. Look at which normals are parallel:All three normals parallel: the planes are three parallel planes, some of which may coincide. Results: a whole plane (all coincident), or nothing.Exactly two normals parallel: if those two planes coincide, the third cuts them in a line. If they are distinct, the third cuts them in two parallel lines, and there is no common point.No two normals parallel: each pair of planes meets in a line, and all three lines are parallel (their directions are all perpendicular to the plane of the normals). Either the three lines are the same line (book pages) or they are three different parallel lines (a triangular prism, with no common point).ConfigurationNormalsCommon pointsThree planes through one pointnot coplanar: n₁ · (n₂ × n₃) ≠ 0one pointThree planes through one line (like pages of a book)coplanar, no two parallela lineTwo coincident planes, third crossing themtwo parallela lineThree coincident planesall parallela whole planeThree parallel, distinct planesall parallelnoneTwo coincident planes, third parallel to themall parallelnoneTwo parallel planes, third crossing bothtwo parallelnone (two parallel lines)Triangular prism: each pair meets in a line, the three lines are parallelcoplanar, no two parallelnoneThe eight configurations of three planes, organized by how they intersect.three parallelno pointstwo parallel + oneno pointsprismno pointscommon linea lineWhen all three normals lie in one plane (n₁ · (n₂ × n₃) = 0), there is a direction parallel to all three planes. Looking straight along that direction, each plane is seen edge-on as a line. These are the four pictures you can get with three different planes. In the last one, the dot is the common line seen end-on.Three planes never meet in exactly two pointsIf two different points P and Q were on all three planes, then the whole line PQ would be on each plane (a plane contains the line through any two of its points). So the common points would be a whole line, not just two points.Up to three lines and planes: the full pictureThe curriculum asks you to organize all configurations of up to three lines and planes by how they intersect. Here is the summary of this unit so far.ObjectsPossible intersectionsTwo lines in two-spaceone point; none (parallel); a line (coincident)Two lines in three-spaceone point; none (parallel or skew); a line (coincident)A line and a planeone point; none (parallel); the whole lineTwo planesa line; none (parallel); a plane (coincident)Three planesone point; a line; a plane; noneSummary of the configurations studied in this unit.Worked exampleExample 1: A single pointShow that the planes x + y + z = 1, x − y + z = 0 and x + 2y − z = 4 meet in exactly one point, and find it.n₂ × n₃ = [1, −1, 1] × [1, 2, −1] = [(−1)(−1) − (1)(2), (1)(1) − (1)(−1), (1)(2) − (−1)(1)] = [−1, 2, 3].n₁ · (n₂ × n₃) = [1, 1, 1] · [−1, 2, 3] = 4 ≠ 0, so there is exactly one point.The first two planes meet in the line (1/2 − t, 1/2, t) (from the lesson Two Planes and Their Line of Intersection).Substitute into the third: (1/2 − t) + 2(1/2) − t = 4, so 3/2 − 2t = 4 and t = −5/4.Point: (1/2 + 5/4, 1/2, −5/4) = (7/4, 1/2, −5/4). Check the third plane: 7/4 + 1 + 5/4 = 4 ✓.They meet only at (7/4, 1/2, −5/4).Worked exampleExample 2: A triangular prismDescribe how the planes x + y = 1, y + z = 1 and x − z = 3 intersect.Normals [1, 1, 0], [0, 1, 1], [1, 0, −1]. No two are parallel.Notice [0, 1, 1] + [1, 0, −1] = [1, 1, 0], so the normals are coplanar and the triple product is 0: there is no single point.Add the second and third equations: x + y = 4. But the first equation says x + y = 1. Contradiction.So no point lies on all three planes. Each pair meets in a line parallel to n₁ × n₂ = [1, −1, 1].No common point: the planes form a triangular prism (three parallel lines of intersection).Worked exampleExample 3: A common lineDescribe how the planes x + y = 1, y + z = 1 and x − z = 0 intersect.The normals are the same as in Example 2, so they are coplanar and no two are parallel.Subtract the third equation from the first: (x + y) − (x − z) = 1 − 0, which is y + z = 1, exactly the second equation. So the equations are consistent and one of them is redundant.Solve the first two: let y = t. Then x = 1 − t and z = 1 − t.Check the third: (1 − t) − (1 − t) = 0 ✓ for every t.All three planes contain the line r = [1, 0, 1] + t[−1, 1, −1] (like pages of a book).Worked exampleExample 4: Two parallel planes and a thirdDescribe how 2x − y + z = 3, 4x − 2y + 2z = 1 and x + y + z = 0 intersect.n₂ = [4, −2, 2] = 2n₁, but 2 × 3 = 6 ≠ 1, so the first two planes are parallel and distinct.The third normal [1, 1, 1] is not parallel to them, so the third plane cuts each of the first two in a line. Those two lines are parallel.No common point: two parallel planes crossed by a third.Just for funThree planes can meet in pairs and still never all meet at once: the two sloping sides of a pup tent and the ground do exactly that. So camping is secretly a lesson in the triangular-prism configuration.Hands-on activityThree-plane galleryYou need: three hardcover books (or three pieces of stiff cardboard), a box, and a table.Build each configuration and say how the three "planes" intersect:One point: the inside corner of a box (floor and two walls).A line: open one book and fan three of its pages; they all share the spine.Prism: lean two books against each other like a tent on the table. The table is the third plane. Check that no single point touches all three.Three parallel planes: stack the three books with a pencil between each pair as a spacer.Two parallel planes and a third crossing them: stack two books with a spacer, then hold the third book upright so it cuts through the gap. It meets each of the two in a line.What to look for: only arrangement 1 has exactly one common point. In 2 the common points form a line. In 3, 4 and 5 there is no point on all three planes, even though in 3 and 5 some pairs do meet.Your turn1How do the planes x = 0, y = 0 and z = 0 intersect?Show answerIn one point, the origin.2How do x + y + z = 1, 2x + 2y + 2z = 2 and 3x + 3y + 3z = 3 intersect?Show answerAll three are the same plane: the intersection is the whole plane x + y + z = 1.3How do x + y + z = 1, x + y + z = 2 and x + y + z = 3 intersect?Show answerThree parallel, distinct planes: no common points.4Three planes have normals [1, 0, 1], [0, 1, 1] and [1, 1, 0]. Do they meet in a single point?Show answer[0, 1, 1] × [1, 1, 0] = [(1)(0) − (1)(1), (1)(1) − (0)(0), (0)(1) − (1)(1)] = [−1, 1, −1]. Then [1, 0, 1] · [−1, 1, −1] = −2 ≠ 0. Yes, one point.5How do x − y = 0, y − z = 0 and x − z = 0 intersect?Show answerThe first two add to x − z = 0, the third: consistent and redundant. Common line: x = y = z, that is, r = t[1, 1, 1].6How do x − y = 0, y − z = 0 and x − z = 5 intersect?Show answerThe first two give x − z = 0, contradicting x − z = 5. No two normals are parallel, so this is a triangular prism: no common point.7Which configuration of three planes is the only one with exactly one common point?Show answerThree planes whose normals are not coplanar (n₁ · (n₂ × n₃) ≠ 0), like the floor and two walls at a corner.More practice: warm-up1Can three planes meet in exactly two points?Show answerNo. Two common points would force a whole common line.2A floor and two walls meet in a corner. What is their intersection?Show answerA point.3Three pages of an open book: what is their intersection?Show answerA line (the spine).4If n₁ · (n₂ × n₃) ≠ 0, what is the intersection?Show answerExactly one point.5How many points do three parallel, distinct planes share?Show answerNone.More practice: on your own1Classify 2x − y + z = 3, 4x − 2y + 2z = 1 and x + y + z = 0.Show answerFirst two are parallel and distinct (normals parallel, 2 × 3 ≠ 1). The third crosses both. No common point, two parallel lines of intersection.2Classify x + 2y − z = 1, 2x + 4y − 2z = 2 and x − y + z = 3, and find the intersection.Show answerThe first two coincide (second = 2 × first). The third is not parallel, so they share a line. Solve x + 2y − z = 1 and x − y + z = 3: adding gives 2x + y = 4. Let x = t: y = 4 − 2t, z = 3 − t + (4 − 2t) = 7 − 3t. r = [0, 4, 7] + t[1, −2, −3]. Check t = 0 in the first: 0 + 8 − 7 = 1 ✓.3Show that P₁: x + y + z = 1, P₂: x − y + z = 0 and P₄: 2x + 2z = 1 share a line.Show answerP₄ = P₁ + P₂ (add the equations), so every point on both P₁ and P₂ is on P₄. Triple product: [1, −1, 1] × [2, 0, 2] = [−2, 0, 2], and [1, 1, 1] · [−2, 0, 2] = 0 ✓. Common line: (1/2 − t, 1/2, t).4Give three planes, none of them a coordinate plane, that meet only at (1, 2, 3).Show answerFor example x = 1, y = 2, z = 3 (or x + y = 3, y + z = 5, x + z = 4; check: 1 + 2 = 3, 2 + 3 = 5, 1 + 3 = 4 ✓, and the triple product is [1, 1, 0] · ([0, 1, 1] × [1, 0, 1]) = [1, 1, 0] · [1, 1, −1] = 2 ≠ 0).5Explain why three planes with no two normals parallel cannot meet in a whole plane.Show answerA common plane would need all three planes to coincide, which needs parallel normals. So the intersection is a point, a line or nothing.Challenge1For which k do x + y + z = 1, x − y + z = 0 and kx + y − z = 2 fail to meet in a single point? Describe that case.Show answern₂ × n₃ = [1, −1, 1] × [k, 1, −1] = [(−1)(−1) − (1)(1), (1)(k) − (1)(−1), (1)(1) − (−1)(k)] = [0, k + 1, k + 1]. n₁ · that = 2k + 2, which is 0 when k = −1. Then the third plane is −x + y − z = 2, that is, x − y + z = −2: parallel to the second plane (x − y + z = 0) and distinct. The first plane crosses both: no common point.2Using P₁: x + y + z = 1 and P₂: x − y + z = 0, find a plane P₃ that meets them in a single point and a plane P₄ that meets them in more than one point.Show answerP₃: z = 0. [1, −1, 1] × [0, 0, 1] = [−1, −1, 0] and [1, 1, 1] · [−1, −1, 0] = −2 ≠ 0, so one point: (1/2, 1/2, 0). P₄: 2x + 2z = 1 (P₁ + P₂), which contains their whole line of intersection. (Other answers are possible.)3Suppose n₁ · (n₂ × n₃) = 0 and no two normals are parallel. Explain why the three lines where pairs of planes meet are all parallel.Show answerThe three normals lie in a common plane; let d be a normal to that plane. Each line of intersection has direction nᵢ × nⱼ, which is perpendicular to both nᵢ and nⱼ, so it is perpendicular to the plane they span: it is parallel to d. All three lines are parallel to d, so they are parallel to each other (or coincide).Check your understandingQuiz1. Three planes meet in exactly one point when:n₁ · (n₂ × n₃) ≠ 0n₁ · (n₂ × n₃) = 0two of the normals are parallelthe normals all lie in one plane2. The floor and two adjacent walls of a room intersect in:a linea planenothinga point3. Each pair of three planes meets in a line, the three lines are parallel and distinct, and no point is on all three. This is called:a sheaf of planes through one linea triangular prismthree coincident planesthree planes through one point4. How do x + y + z = 1, x + y + z = 2 and x − y = 0 intersect?one pointa lineno common pointa plane5. Which of these can NEVER be the set of common points of three planes?exactly one pointa lineno pointsexactly two points6. How do the planes x − y = 0, y − z = 0 and x − z = 0 intersect?only at the originin no pointin the plane x = yin the line r = t[1, 1, 1]Mark lesson completeFor parents and teachersBig idea (C3.3, C4.3): three planes meet in a point, a line, a plane or nothing, in eight possible arrangements. The triple product of the normals is the quick test: non-zero means exactly one point; zero means one of the other seven arrangements, which you sort out by checking which normals are parallel and whether the equations are consistent.How to help: Build the configurations with books and a box (the activity). Your teen should be able to point to each one and say "point", "line", "plane" or "nothing". Ask "Which pairs of planes are parallel?" before any algebra.Common difficulties:Thinking "no common point" means the planes are parallel. In a prism, no two planes are parallel.Thinking three planes meeting in pairs must have a common point.Triple-product arithmetic. Check by noticing simple relations, such as one normal being the sum of the others.Forgetting that coplanar normals still allow a common line if the equations are consistent.PreviousIntersection of a Line and a PlaneNextSolving Systems of Three Planes Algebraically