Is a line, that's contained in a plane, parallel to it?

Question

Have got into a pretty heated debate with a friend, and looking online there's lacking proof Line contained in a plane

Is a line that is contained within a plane, considered parallel to it? By my understanding it is parallel , if at all points the line has equal distance to the plane, which in this case is always 0, but they are debating that it intersects the plane at infinitely many points, so it cannot be considered parallel, is there some mathematical proof that would prove or disprove either statement?

I see there is heated debate online about whether a line that matches another line is parallel to it and I'm wondering if there's a clean answer about this.

Answer 1

I assume you are talking about planes and lines in $\mathbb{R^3}$. If so, then all planes in are defined by a normal vector and a point, while all lines are defined by a direction vector and a point. It follows that some line will be parallel to some plane if that line's direction vector is perpendicular to that plane's normal vector, which reduces to checking their dot product.

P.S. This is just a preliminary answer. Do let me know if you require a more in-depth one and I will write it for you.

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Answer 2

In Euclidean space for two lines to be parallel they must never intersect. Since a line in a plane intersects the plane it cannot be parallel to it. Note that the lack of intersection is necessary but not sufficient in higher dimensions.

Answer 3

This is entirely a question of convention. I'd say yes, because I like "parallelness" to be an equivalence relation, but it frankly does not matter in the slightest, providing you're clear about how you are using the word.

Answer 4

This is a matter of opinion and convention, and one definition may be more useful than another definition for some purposes. This is common in mathematics.

One definition of parallel lines in a Euclidean plane is that they are equidistant from each other. This breaks down in the context of the elliptic or hyperbolic plane and it depends on a distance metric. Perhaps a better definition is that they do not intersect. Again, this breaks down in the context of the elliptic or hyperbolic plane. However, in the context of the projective plane, this must be modified to state that parallel lines intersect only at one point at infinity.

Moving up to three dimensional space, there is the concept of skew lines which don't intersect each other, and are not parallel since they are not in the same plane. Introducing a plane at infinity, the definition of parallel lines is the same as in a fixed plane. That is, they only intersect at one point at infinity. Any plane intersects the plane at infinity in a line at infinity. Two parallel planes only intersect at the same line at infinity.

It now makes sense that each line in a fixed plane intersects the same line at infinity as the fixed plane does and exactly at one point. This also holds for each line in a plane parallel to the fixed plane. Thus, there are good reasons to consider every line in a plane to be parallel to that plane.

Answer 5

I'll throw my hat into the ring.

I certainly agree that it is a matter of convention.

However, it is also reasonable to presume that 3 dimensional Geometry is (in effect) an extension of 2 dimensional Geometry.

In 2 dimensional Geometry, I have never heard of the specification that a line is parallel to itself. This (inconclusively) suggests that in 3 dimensional Geometry, a line can not be considered parallel to the plane that contains the line.

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This is where the fun starts. Normally, definitions, like the concept of parallel lines in 2 dimensional Geometry, are created in order to facilitate solving problems. So, a hidden issue is the question of why the concept, in 3 dimensional Geometry, of a line being parallel to a plane, was created in the first place.

That is, what math problems is the definition designed to help solve. Having identified these problems, you would then ask which convention facilitates solving these problems, whatever they are.