I was helping a high school student with some fundamental concepts of planes and lines, when I realized I am rusty on some definitions myself. I found some minimal coverage in his high school math and precalculus textbook, and I also found a section on my vector calculus text by Marsden. However, none of these coverage seems to be comprehensive - in particular, I could not even find a definition of what it means for two planes to be perpendicular.

Geometry books like Hilbert and Hartshorne are not at the right level for a senior high student or a freshman. I would imagine there are some books out there that lays out the fundamental definitions, theorems, and proofs involving the perpendicularity of lines and planes, and their parametric and symmetric equations.

What kind of books cover these things now? Or are students expected to learn this from the Internet?

Note: I'm looking for something that can be a reference for me and the student. I think these linear algebra notes are perfectly reasonable level-wise. I was not able to find a definition for "perpendicular planes" in my linear algebra text or these notes, however. In the linear algebra course I took a while back, "orthogonal complements" and "orthogonal subspaces" are covered, but these two are distinct from the concept of "perpendicular planes".

  • $\begingroup$ At what level are search results such as duckduckgo.com/?q=linear+algebra+lecture+notes for you? Too advanced? Are you looking for references for yourself or the student? What do you already know? $\endgroup$
    – Tommi
    Commented Nov 30, 2015 at 14:57
  • $\begingroup$ What country or state are you in? I ask because curricula are not global. In particular, in most states there is a stand-alone high school geometry course, but not in all. $\endgroup$
    – mweiss
    Commented Nov 30, 2015 at 15:33
  • 1
    $\begingroup$ @TommiBrander Thanks for the link. I'm looking for something that can be a reference for me and the student. I think these linear algebra notes are perfectly reasonable level-wise. I was not able to find a definition for "perpendicular planes" in my linear algebra text or these notes, however. In the linear algebra course I took a while back, "orthogonal complements" and "orthogonal subspaces" are covered, but these two are distinct from the concept of "perpendicular planes". $\endgroup$
    – FreshAir
    Commented Nov 30, 2015 at 23:57
  • $\begingroup$ @mweiss The student is in Michigan. $\endgroup$
    – FreshAir
    Commented Nov 30, 2015 at 23:57
  • 1
    $\begingroup$ This isn't an answer to the question as posed, but symbolic variables, indexed lists with three entries, a dot product, and a notion of a right angle should be sufficient to give an elementary definition. If you include addition of points and additive inverses, you can define planes that don't pass through the origin to your discussion. If you include lists of lists and indexed lists with four entries, you can nicely define some very interesting maps between such objects. Observing and exploring their properties would be good for a student at this level. $\endgroup$
    – Andrew
    Commented Dec 19, 2015 at 4:22

2 Answers 2


If the student attends a school that has a distinct Geometry course (typically taken in 9th or 10th grade) then it will probably use a textbook similar to Glencoe Geometry or HMH Geometry. I have one of those open in front of me now; looking at it, I see that the sequence of topics runs like this:

  1. First, "perpendicular lines" are defined as lines that intersect to form 90° angles.
  2. Second, "a line perpendicular to a plane" is defined to mean a line that is perpendicular to every line in the indicated plane.
  3. "Parallel planes" are defined, and one of the postulates in the book is that if two planes are non-parallel then they intersect in a single line.

But (somewhat to my surprise) "two perpendicular planes" does not appear to be defined or ever used anywhere in the context of the book. I don't know that it's possible to say why that concept is never invoked, but I have a conjecture: The typical (?) definition of "perpendicular planes" goes like this:

A planes $M$ is perpendicular to a plane $N$ if there exists some line $m$ in $M$ that it perpendicular to every line $n$ in $N$.

That, at least, is the definition I found in the first few Google search results for "perpendicular planes definition" (examples: here and here). Notice, first, that this definition involves two quantifiers: An existential quantifier on $m$ (we only need to find one line $m$ in $M$ that meets the required condition), and a universal quantifier on $n$ (once we have chosen $m$, every line $n$ in $N$ needs to be perpendicular to it). That's conceptually quite subtle, arguably on a par with the well-known conceptual difficulties around the quantification in epsilon-delta proofs.

Second, notice that although the property being defined is symmetric (i.e. "$M$ is perpendicular to $N$" means exactly the same thing as "$N$ is perpendicular to $M$"), the definition is not. In fact it is not at all obvious that the existence of an $m$ in $M$ that is perpendicular to every $n$ in $N$ implies the existence of a specific $n$ in $N$ that is perpendicular to every $m$ in $M$.

I suspect that the conventional wisdom among curriculum designers, to the extent that it has given any thought to the matter at all, is that the potential benefit of defining perpendicular planes is outweighed by the difficulties entailed by the definition.

It seems to me that another possible definition would be to say:

Suppose planes $M$ and $N$ intersect in line $l$, and choose any point $P$ on $l$. Let $m$ be the line in $M$ that is perpendicular to $l$ at $P$, and let $n$ be the line in $N$ that is perpendicular to $l$ at $P$. Then planes $M$ and $N$ are called perpendicular if the lines $m$ and $n$ are perpendicular.

This definition has the advantage of being obviously symmetric in $M$ and $N$. It also generalizes, with very little work, to define a dihedral angle: The angle between $M$ and $N$ is defined to be the angle between $m$ and $n$. With this definition, $M$ is perpendicular to $N$ if the dihedral angle between them is 90°.

But whatever advantages this other definition may have, they are also probably outweighted by the semantic complexity of the definition, which requires six geometric objects (not only the two planes $M$ and $N$ but also the three lines $l, m$ and $n$ and the point $P$) to define a single relationship. For 9th or 10th graders, it's a disaster waiting to happen.

So it does not surprise me that Geometry textbooks avoid the problem of defining what it means for two planes to be perpendicular. On the other hand it does surprise me that they avoid the concept altogether. High school geometry typically treats rigor as a kind of aspirational value, not an absolute commitment. I think it's pretty surprising that I don't find any exercises that show, for example, a three-dimensional solid (say a frustrum) and ask the student to identify which planes are parallel, which are perpendicular, and which are neither.

Now, all of the above assumes you are interested in the geometry of lines and planes in space from a synthetic or Euclidean perspective. But if you want "parametric and symmetric equations" for planes (which I see was part of your question), then a Geometry course is the wrong place to look -- you want to look at a Precalc book. I have one in front of me: It's the Glencoe Advanced Mathematical Concepts: Precalculus with Applications. In this textbook we find that:

  • An equation of the form $ax+bc+cz=d$ defines a plane in 3-dimensional space
  • So a system consisting of two such equations defines two planes, which either are parallel (in which case there is no solution), or intersect in a single line (in which case there are infinitely many solutions that can all be expressed in terms of a single parameter), or coincide (in which case there are infinitely many solutions that can all be expressed in terms of two parameters)
  • And a system consisting of three such equations defines three planes, which either do not intersect, or intersect in a line, or intersect in a single point, or coincide.

A typical treatment looks something like this: enter image description here

But notice that in this context -- solving systems of equations in three variables -- it does not matter whether the planes are perpendicular or not. All that matters is whether they intersect; the angle of intersection is irrelevant.

Much later in a Precalc book, we encounter information about vectors, dot products, and cross products. In this context students learn that two vectors are perpendicular if and only if their dot product is zero. They also learn that the cross product of two vectors is perpendicular to the plane in which those vectors lie:

enter image description here

Such properties make it possible for students to answer questions like:

  • Given two vectors, determine if they are perpendicular
  • Given three points in space, determine if any of the vectors joining those points are perpendicular to each other
  • Given two vectors, find a third vector perpendicular to both
  • Given three points in space, find a vector perpendicular to the plane containing the three points

Students also learn to:

  • Given a point $P$ and a vector $\vec{v}$, write a pair of equations for the line through $P$ in the direction of $\vec{v}$ in parametric form
  • and vice versa.

This all gets pretty close to what you are asking for. But I still don't see in this textbook any place where students encounter the definition of a normal vector to a plane, or study the relationship between the equation of a plane and its orientation in space, or talk about what it means for two planes to be perpendicular. I think that material is simply beyond the scope of high school mathematics in the United States.

One more thought: You asked about "Fundamental definitions, theorems, and proofs". Quite apart from content, you should probably be aware that very few mathematics textbooks at the secondary level (with the exception of Geometry!) are organized along "definitions, theorems, and proofs". Students learn techniques for solving problems, but those techniques are not organized into a logico-deductive structure, in which some properties are taken as primitive and others derived. That is not to say that such a structure does not exist at all, but it is not made explicit; the word "Theorem" is almost never used, and when it is used, normally a Theorem is merely illustrated, not proved.

  • $\begingroup$ Thank you very much, mweiss, for this detailed answer. $\endgroup$
    – FreshAir
    Commented Apr 11, 2016 at 6:50

In Finland this is covered (very briefly) in gymnasium, in the third course of extended mathematics, so most such course books should cover the subject. These are not rigorous treatments.

The one I'm looking at right now (Pyramidi 3) does the following:

  1. Fail to define the angle between two lines. One exercise does concern this, but it is left up to the teacher to mention. This is not emphasized at all in the text.
  2. Define the normal (line) to a plane. The angle between the normal and the plane is 90 degrees.
  3. Define the angle between a line and a plane (using a projection, which I doubt has been defined anywhere, so it is left as a heuristic throw-away).
  4. Through any point outside the plane there is exactly one parallel plane.
  5. The angle between parallel planes is 0 degrees.
  6. The angle between two intersecting planes is the angle between their normals.

There is one example and a total of 3 exercises, after which the subject changes. Everything takes two spreads (4 pages, page size somewhat smaller than A4): one spread for the definitions and one for the example and the exercises and one huge picture of a book that relates to one exercise but is not essential.

I don't remember explicit treatment of this in the university either, but might be that I have forgot. Linear algebra courses focused on other matters.

  • $\begingroup$ Thank you Tommy Brander. $\endgroup$
    – FreshAir
    Commented Apr 11, 2016 at 6:50

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