addition of bound vectors and usefulness of the bound vector concept

Question

I am trying to explain vectors to a 7th grade student and I am not sure that the concept of bound vectors is really useful. I'd rather just go straight to free vectors. For instance, I understand that addition is only defined for free vectors and not for bound vectors. Would you agree with this statement?

So I am not convinced that bound vectors are a useful concept. The only possible application I can imagine is when talking about "position vectors" but I 'd rather explain what they are without using the "bound vector" concept. In short, I am not sure what's the most reasonable and helpful way to introduce bound, position and free vectors (and in what order). The way I see it, both free and position vectors can be defined with only two numbers in the Cartesian plane, while bound vectors need four numbers. Given that the algebraic treatment of vectors in 2D uses two numbers, this further suggests that bound vectors do not provide any helpful insights to students.

Any thoughts on the subject and, in particular, is the statement "addition is not defined for bound vectors" correct? Obviously when I say "defined" I mean in the general case, for any two bound vectors.

Answer 1

You are asking for an opinion. The Wikipedia article Euclidean vector distinguishes between "free" and "bound" vectors. Addition can be defined on bound vectors with the same origin using the parallelogram law. Another addition is like composition of maps in Category theory which requires terminal point to match initial point of the second vector.

However, it is possible to define a general addition for any two bound vectors. Given $AB$ and $CD$, two bound vectors, let $E$ and $F$ be the two midpoints of the two vectors and $G$ be the midpoint of $EF$. Now let $GP := GA+GC,\; GQ := GB+GD,\;$ and define $AB+CD:=PQ.$ Actually, this is more like a weighted sum as in affine space. So, more accurately, $\frac12AB+\frac12CD=PQ.$

The situation is somewhat similar to a Groupoid except that every point in space is an object of the category, with only the identity selfmap, and a bound vector is the unique map from the initial point to the terminal point.

If you don't find bound vectors useful, then fine, but they arise in physics, engineering, and geometry for good reasons.

Answer 2

I have never seen the term "bound vector". The important thing is to distinguish between vectors and positions, since they look alike on paper.