# Defining/introducing vectors informally

A vector is a collection of elements whose order matters. If the elements represent comparable measurements it can be said that they together represent a direction and a magnitude (length). The Pythagorean theorem is used to find the length of a vector.

This is to be represented to high school students. I want to ask for opinions so the definition can be as useful as possible while not misleading them.

EDIT: forgot to mention that the students are familiar with what a set is.

• In high school, the term "vector" should be used in the sense of physics. Not in the sense of computer science as you wrote in your first sentence. If you want the computer science meaning, call it a "list" or "string" something. – Gerald Edgar Mar 30 '16 at 15:40
• @GeraldEdgar Is it really that straightforward that they "should"? Can I ask you to layout your reasoning? – snoram Mar 30 '16 at 16:36
• See the mweiss answer for reasons. – Gerald Edgar Mar 31 '16 at 0:47
• @mweiss It is a member of the vector space {banana}x{horse}x$\mathbb{R}$. – Jessica B Mar 31 '16 at 11:15
• @mweiss Over $\mathbb{R}$. It's naturally isomorphic to $\{*\}\times\{*\}\times\mathbb{R}$. I could come up with some alternative vector spaces if you prefer, say over a finite field. – Jessica B Mar 31 '16 at 15:20

I think the "ordered $n$-tuple" notion of "vector" is not only unnecessarily abstract, it actually distorts the meaning.

I informally define a vector as directed line segment (or "arrow") that "floats" in space. A vector has magnitude and direction, but no specific position. Crucially, the components of a vector are only defined relative to a coordinate system, and sometimes it is helpful to choose different coordinate systems for different problems. For an objects sliding down a ramp, the "best" coordinates are parallel to and perpendicular to the ramp; for an object flying through the air, the "best" coordinates are horizontal and perpendicular. (Sometimes you may even change coordinates in the middle of a problem!) So a single vector may have more than one representation as an ordered list.

• My I ask: Why the downvote? – mweiss Mar 30 '16 at 20:02
• "Floating" is an informal way of expressing the idea of an equivalence class. More formally, we can begin with the set of all directed line segments ("arrows"), and then define two such arrows to be equivalent if they have the same magnitude and direction. A vector is then an equivalence class of arrows. But the OP specifically asks for an informal definition, so we use the (informal) notion of "floating" to capture the essence of the idea. – mweiss Mar 31 '16 at 2:16
• And vectors defined this way are a precise example of "elements in a vector space": one defines scalar multiplication of arrows as a dilation, and defines addition of arrows via the parallelogram rule. – mweiss Mar 31 '16 at 2:18
• @Andrew My understanding is that downvotes should be for poor quality answers, whereas if you disagree with an answer you should instead upvote one or more you do agree with. – Jessica B Mar 31 '16 at 7:20
• @Andrew: I (kindly) take outrage that you assert that firmly that a vector is an algebraic object. I assert with as much confidence that a vector is a geometric object, and that the algebraic point of view on vectors mainly shows that algebra is convenient to represent geometry. – Benoît Kloeckner Apr 3 '16 at 12:12

The first part of your definition is strange and unclear. What is the meaning of "whose order matters?" The second sentence is much better for a high-school understanding: a quantity with magnitude and direction.

When I teach vectors in my 12th grade Physics class, I do this:

• State the "quantity with magnitude and direction" definition.
• Show a YouTube clip that gives the same definition. (Most of my students are already familiar with this clip.)
• State that the usual way to represent a vector is an arrow whose length and direction matter but whose starting point does not matter. I use actual arrows (which may not be allowed in your school) and slide them around the white board to show equivalent and non-equivalent vectors.
• Discuss real-world examples of vectors such as displacement, velocity, and force.
• Explain how to show a vector variable: with bold-face if in print ($\mathbf v$), with an over-arrow if hand-written ($\vec v$).

All that can be done pretty quickly. I then take more time to discuss vector operations, such as addition, negation, subtraction, scalar multiplication, decomposition into components, and re-composition from components. I briefly state that other operations exist (e.g. dot and cross product) but we will not discuss them further in my class. I also state that other, more advanced and complicated definitions of vectors exist but that my definition suffices for the class.

This works pretty well, with no resulting confusion about vectors. The difficulty comes in their application, a little later in the chapter. None of my students who do further math and science in college has ever told me that they were confused or mislead by my treatment of vectors, and I do ask about such things.

• Brought in that "order matters" to contrast sets. – snoram Mar 30 '16 at 12:53
• @snoram: I see, so you mean an ordered n-tuple. I avoid that definition in Physics and have never needed it in other classes. It works for me, but I can see the possible need for n-tuples in other situations. I also stick to one- and two-dimensional vectors, briefly stating that higher dimensions are also possible. – Rory Daulton Mar 30 '16 at 12:57
• Personally, I always really disliked this "floating arrow" definition. I much prefered the tuple definition, and still do. In later math, it seems that "floating arrow" only makes sense in an affine space. On a manifold, you will need a connection to parallel translate your vector, and vectors are more primitive than parallel translation. Even in basic mathematics, we often want to analyze "nongeometric" vectors (like (#of cars,# o passangers)) for which the tuples definition makes much more sense than the geometric. – Steven Gubkin Mar 30 '16 at 17:34
• @snoram I formally met vectors at school long before sets (at university). – Jessica B Mar 31 '16 at 7:18
• @snoram Ok, I just spotted the edit. – Jessica B Mar 31 '16 at 7:34

I think the elements of your suggested definition are ok, but I would want to split them up and put a lot more explanation in between.

I would perhaps start by saying that there are different ways of thinking about vectors, and that while they are all the same in some sense (and so it's reasonable to use the same word), it might not be obvious to everyone until they get used to switching the way they think about them.

Then, according to time/syllabus, I'd introduce the different ways of looking at vectors in the contexts that each is used: position vectors in coordinate geometry, free-floating arrows in coordinate-free geometry, direction/magnitude in mechanics, $n$-tuple of objects in sequences or programming.

For each one I'd get them doing questions in that context. After that, I'd move on to questions that involve changing between two of the descriptions, eg Cartesian and polar coordinates. Doing such questions should allow students to mentally connect the definitions, although some won't manage that level of understanding.

The reason I think your definition is potentially confusing is that it mixes the different viewpoints, at a point when the students don't have the understanding to see the connections. If you're just working with $n$-tuples, direction and length don't have any meaning, particularly if the entries might not be real numbers, and making the shift in thinking is likely to take longer than reading one sentence. Using Pythagoras only makes sense when you have some kind of coordinates, and seeing why it has anything to do with something other than triangles will take time too. The students need to get their heads around the different forms before they can put the different sentences together, if they are to be happy with a definition presented that way.