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Why can a line never intersect a plane in exactly two points?

I know this seems like a really simple question, but I'm having a hard time figuring out how to answer it. I also tried googling the question but I couldn't find an answer for exactly what I'm looking for.

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    $\begingroup$ Fact $1$: If a line passes through $2$ points of the plane, then it is included in the plane. Admitting fact $1$, the answer to your question is clear. Now how to prove fact $1$? $\endgroup$ – Leaning Jun 16 at 20:39
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    $\begingroup$ Common sense approach: take a straight piece of paper, assume it represents a plane. Take a pencil, assume it represents a line. Since the line is not "curved", it can intersect with the plane in 1 point, but for it to hit another point in the place, either the line needs to bend or the plane needs to bend. $\endgroup$ – NoChance Jun 16 at 21:20
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    $\begingroup$ @NoChance: I used my "common sense" in elementary school to deduce that it's quite possible to draw 2 lines connecting 2 points. That's how I realized my sense wasn't that common :-) $\endgroup$ – Mehrdad Jun 17 at 5:30
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    $\begingroup$ One of Hilbert's axioms is If two points $A$, $B$ of a line $a$ lie in a plane $α$, then every point of $a$ lies in $α$. In this case we say: "The line $a$ lies in the plane $α$". $\endgroup$ – Henry Jun 17 at 7:07
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    $\begingroup$ You have a lot of answers now. Can you perhaps use them to clarify "exactly what you're looking for"? As it stands, your question is somewhat vague about the nature of the desired exactitude. $\endgroup$ – Lee Mosher Jun 17 at 14:08

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If you pick two points on a plane and connect them with a straight line then every point on the line will be on the plane.

Given two points there is only one line passing those points.

Thus if two points of a line intersect a plane then all points of the line are on the plane.

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    $\begingroup$ "then every point on the line will be on the plane." this feels a bit hand-wavy $\endgroup$ – MichaelChirico Jun 19 at 6:09
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    $\begingroup$ Why is it hand wavy? It’s geometry. A plane is flat and infinitely large with no thickness. If two points that exist on it connect, all other points between them connect because there is no possibility of leaving the plane and rejoining it. $\endgroup$ – vol7ron Jun 19 at 11:30
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I think you're having trouble with the question because there isn't a satisfying answer. The statement "a line can never intersect a plane at exactly two points" is either an axiom in some formalization of Euclidean geometry or follows so directly from one or two other axioms in the system that the answer seems empty of meaning, a restatement of definitions (as in some of the good answers here).

An axiom is a statement that's taken as a given, and that's where mathematics starts. The question of why any particular axiom exists or is justified is to some extent a philosophy of science question. In the case of Euclidean geometry I think the answer is that the rules seem to (mostly) mirror our experience of the physical world we inhabit and the mathematical results of the system lead to useful practical results (helpful in building a house, for instance).

But there are other systems of geometry with different/fewer axioms which seem intuitively absurd yet produce useful results as well.

EDIT: see Paul Sinclair's answer

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As jberryman has expressed, the real issue here is what is meant by a "line" and a "plane". Traditionally, these are taken as undefined primitive concepts, and the ideas that any two points will determine a unique line, and that if two points are in a plane, then the line through those two points will lie entirely in that plane are taken as axioms.

The problem with definitions is that they can only introduce new ideas in terms of older ideas. But you have to start somewhere. The base terminology of a theory, such as geometry, cannot be defined in a normal fashion. Instead, we call these terms "primitives", and choose certain statements, called "axioms" or "postulates" relating them to each other which we simply assert to be true. It is this list of primitives and axioms that establish what the theory we are working on is. In a sense, the axioms define the primitives, by establishing how they relate to each other.

While it is common to treat "line" and "plane" as primitives, it is not absolutely necessary. You can build up all of Euclidean geometry just from the primitive term "point" and the primitive relationship "these two points are closer together than those two points", and basic set theory. (It is not wise to do so, as it requires a lot of niggling axioms to make sure that these primitives encompass the desired behavior, which would thoroughly confuse new students of the subject.)

The general outline of this approach is:

  • Define distance by equivalence classes on the relation "$A$ and $B$ are neither closer together nor farther apart than $C$ and $D$"
  • Define "$B$ is between $A$ and $C$" if among all points $D$ the same distance from $B$ as $C$, $C$ is the farthest from $A$.
  • Define "$A,B,C$ are colinear" if one is between the other two.
  • Define a set of points to be "linearly closed" if for any two points $A, B$ in the set, if $C$ is colinear with them, then $C$ is also in the set.
  • A "line" is a linearly closed set where every trio of points are colinear.
  • The "span" of a set of points is the smallest linearly closed set of points containing it.
  • A "plane" is the span of some set of three non-colinear points.

With this system, it is by definition that once a line and a plane share two points, the line must lie in the plane, as every point on the line must be colinear with those two points, and therefore must be in the plane, as the plane is linearly closed.

Whether you choose to do some grandiose scheme of definitions as above, or take the simple route of having "line" and "plane" as primitives and make this condition an axiom, it is something you want to have in your geometry, because it expresses the idea of what we want a plane to be: something that expands the idea of "straightness" into another direction.

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    $\begingroup$ “…the real issue here is what is meant by a "line" and a "plane". Traditionally, these are taken as undefined primitive concepts…” This is traditional in axiomatic Euclidean geometry and similar approaches, and your answer is good with respect to these. But those axiomatic approaches are aiming to axiomatise a pre-existing notion of space — either informally/intuitively, or modelled mathematically as e.g. $\mathbb{R}^3$, or something along similar Cartesian lines — and that sort of non-axiomatic view is just as important for this question. $\endgroup$ – Peter LeFanu Lumsdaine Jun 18 at 11:27
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    $\begingroup$ @PeterLeFanuLumsdaine - Note that this forum is called :"Mathematics", not "Philosophy" or "Sophistry". Proof by logical deduction from axioms is what defines "mathematics". We choose those axioms to support our physical intuitions, but it isn't mathematics until those axioms and primitives are chosen. A discussion of why we want these axioms (or equivalent) is outside the realm of mathematics, which will as happily accept a set of axioms that defy all our intuitions, as long as they are internally consistent. $\endgroup$ – Paul Sinclair Jun 18 at 17:29
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    $\begingroup$ I’m a mathematician myself; I’m well aware of what mathematics is. If I may try my points again: First, there are other rigorous mathematical ways to develop geometry, besides direct axiomatisations of geometry itself — most importantly, defining $\mathbb{R}^3$ within some foundational axiomatic system such as ZFC set theory, and investigating its properties in that system. Indeed, I’m pretty sure most mathematicians today think of 3-dimensional geometry primarily this way, not as through Euclid’s axioms. [cont’d] $\endgroup$ – Peter LeFanu Lumsdaine Jun 18 at 18:16
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    $\begingroup$ Secondly, the connection between axiomatisations and the situations they’re meant to model is absolutely part of the domain of mathematics, and especially an important part of mathematical pedagogy. Hilbert’s “tables, chairs, and beer-mugs” quote is a very important point about the validity of formal proofs, but interpreting it as saying that axiomatic systems are totally arbitrary and intuition is irrelevant is a very extreme formalist position, which I think few mathematicians hold, and which is rather difficult to defend. $\endgroup$ – Peter LeFanu Lumsdaine Jun 18 at 18:21
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    $\begingroup$ @PeterLeFanuLumsdaine - That is a gross distortion of what I said. There is a big difference between "systems that violate our intuition are still valid subjects for mathematical study" and "intuition is irrelevant". Theories that formalize our intuitive concepts are obviously important for mathematics. But theories that violate those concepts have much to teach us as well. Mathematics is big enough to include both. And I still hold that math occurs once the theory is defined. How to define the theory is part of whatever field the theory is modelling. It is vital, but it is not mathematics. $\endgroup$ – Paul Sinclair Jun 18 at 21:59
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Equation of line: $\frac{x-a}{l}=\frac{y-b}{m}=\frac{z-c}{n}$ ...(1)

Equation of plane: $\alpha x+\beta y+\gamma z+\partial=0$...(2)

For intersection, any point on line $P(tl+a,tm+b,tn+c)$ must lie on plane too. Plugging this point in equation of plane will give a linear equation in $t$ which may give exactly one $t\in\mathbb R$ (intersection is a single point) OR an identity $0=0$ true for all $t\in \mathbb R$ (intersection is a line).

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A plane is a convex set. A convex (i.e. linear) combination of any two points in a convex set will be inside the set. So if you have two distinct points in the plane, you automatically have infinite points inside the plane.

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    $\begingroup$ I find the wording “a convex (i.e. linear) combination” rather unfortunate. $\endgroup$ – leftaroundabout Jun 17 at 14:57
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Join the two points on the plane: that's a straight line which we'll call $\;L\;$ and which , btw, is wholly contained in the plane . If another line, say $\;M\;$, intersects those two points on the plane then it intersects $\;L\;$ in those two points. That cannot be, of course, unless $\;L=M\;$ ....

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Think of it this way: We define a line-segment to be the shortest path in space between two points, and a straight line is the union of all line segments between any two of its own points. Then if a straight line intersected a plane in only two points, the segment between those points in the plane would be shorter than the segment between those two points in the line, contradicting the assumption that it is in fact a straight line.

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If points A and B are in the intersection, then the midpoint of the line segment AB is also in the intersection, making three points (assuming A and B are distinct points). This can be continued infinitely, showing that the number of points in the intersection is infinite.

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The difficulty in proving this comes from the fact that whether or not a line, not on a plane, can intersect the plane in more than one place is equivalent to Euclid’s 5th postulate.

So in geometries where two parallel lines may intersect more than once, a line may also pass through a plane more than once.

Example: Consider two lines in a double elliptic geometry such as spherical geometry such that they intersect twice. If you place a plane such that one of the lines lies in the plane, then the other line will pass through the plane twice also.

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First, some definitions:

  • A line is straight and infinitely long.
  • A plane is flat and infinitely long and wide.
  • An intersect is a point shared by the line and the plane.

This leads to three relations between a line and a plane:

  • It is parallel and not part of the plane. This means it has 0 intersects.
  • It is parallel and part of the plane. This means it has infinite intersects.
  • It is not parallel. This means at some point it intersects exactly once.

To make a line intersect the same plane twice it can't be parallel, and either the line or the plane must be curved.

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  • $\begingroup$ Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax. $\endgroup$ – dantopa Jun 17 at 14:37
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    $\begingroup$ What is your definition of "straight"? of "long"? of "wide"? Of "infinite"? In mathematics, you can only define things in terms of things previously defined, or a few primitive concepts that are "defined" by the axioms of the theory (which govern how these primitive concepts interact with each other). Also, how do you know that your three relations are the only ones possible? That is the gist of this question. $\endgroup$ – Paul Sinclair Jun 17 at 16:23
  • $\begingroup$ @PaulSinclair I thought the ideas were insufficiently simplified in the other answers, but your comment and the downvotes make it clear I don't know enough about the subject to add value, I Dunning-Krugered. My apologies. $\endgroup$ – Regret Jun 20 at 9:43
  • $\begingroup$ I didn't downvote you. I don't consider a post worthy of downvotes unless it is egregious in some way (adds confusion, actively misleads, is not related to the question, etc.) I do not think yours was helpful for the reasons mentioned, but it was honest and not harmful, either. I just wanted to point out what needed improved. I'm sorry if this encouraged others to downvote, but I still think it was appropriate to point out. $\endgroup$ – Paul Sinclair Jun 20 at 16:22
  • $\begingroup$ @PaulSinclair I appreciate the feedback. $\endgroup$ – Regret Jun 27 at 12:51
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"Why can a line never intersect a plane in exactly two points?"

Well it can ... if the plane ins't flat or the line isn't straight.

If both the plane is flat and the line straight then it can't.

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  • $\begingroup$ The formal definition of a plane is that it is "flat." The formal definition of a line is that it is "straight." The possibilities you are describing involve a surface and a curve, two mathematical objects with formal definitions that allow for curvature. $\endgroup$ – NoseKnowsAll Jun 19 at 1:27
  • $\begingroup$ So I upvoted this because I believe it should have sat at the bottom with zero votes until we know whether or not this is an Euclidian geometry question. See Sphereland for this line of thinking. $\endgroup$ – Joshua Jun 19 at 18:16
  • $\begingroup$ @Joshua Sorry Joshua. That would require people being kind. $\endgroup$ – Randy Zeitman Jun 19 at 20:42

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