Vectors and directed line segments

Question

A vector is a quantity that has magnitude and direction.

A directed line segment is a line segment that has both a starting and an end point ,so it has a direction

From these definitions, a directed line segment and a vector are different, while some textbooks may treat them interchangeably, meaning that they are synonymous, but they are not the same. I can explain this from my perspective as follows:

A directed line segment is a geometrical object, a set of points that not only has length but also direction.

While vectors may be considered as a physical concept, or abstract mathematical object.

Accordingly, some notations really confuse me: Considering $\overrightarrow{AB}$ and $\overrightarrow{BC}$ two directed line segments Then

$\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$

What really confuses me is how are we adding directed line segments? We can add vectors, but not segments.

We don't add normal line segments, we add their lengths.

Can someone explain this to me if I am getting something wrong?

Answer 1

I agree, in general adding directed segments is not meaningful. Considering a directed segment as a vector with a starting point, we can add the vectors but would not know what starting point assign to the result. To expand on @pranavB23 comment, one can find a sense for this when the starting point of the second segment is the end point of the first, where it's quite natural to state that the starting point of the resulting segment is just $A$, and this is the $\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$ case you have shown.

Actually, this addiction of directed segments is in my experience used only to give students a graphical way of understanding/calculating euclidean vector sum when they are first approaching it, since after all when I draw a vector I'm actually drawing a directed segment. What your book/notes probably wanted to say is something like:

In order to sum vectors $\vec v$ and $\vec u$, we can pick a directed segment starting in any point $A$ we like (usually the origin) with same length and direction as $\vec v$, call it $\overrightarrow{AB}$, and another segment starting at $B$ with same length and direction as $\vec u$, $\overrightarrow{BC}$. Then the resulting $\vec v + \vec u$ has same lenght and direction as $\overrightarrow{AC}$.

One may also think about the matter in this terms: calling $\vec P$ the vector giving the position of a point $P$ in space with respect to any fixed origin $O$, then the vector represented by $\overrightarrow{AB}$ is $\vec v = \vec B - \vec A$, so clearly $\vec v + \vec u = (\vec B - \vec A) + (\vec C - \vec B) = \vec C - \vec A$.

Answer 2

"A quantity that has magnitude and direction" may be an acceptable colloquial description of a vector, but cannot serve as a definition.

"A directed segment", AKA "an ordered pair of points", is already better, but we are not yet there. A vector in $3$D elementary geometry terms is an equivalence class of such pairs, whereby two pairs $(A,B)$ and $(C,D)$ are considered equivalent, if there is a translation $T$ of space such that $T(A)=C$, $\>T(B)=D$.

Now these equivalence classes can be added by means of the well known parallelogram construction. One has to prove that the sum $\vec a+\vec b$ of two vectors is well defined, and then that it has the (geometric) properties that make the set of equivalence classes into a real vector space.

Answer 3

Actually it is possible to define a partial addition operation on the set of directed line segments by restricting the arguments to compatible segments such that the terminal point of the first matches the initial of the second. For these restricted pairs the sum can be defined by the triangle law $\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}$.

It is then not too difficult to show that the partial operation extends to a well-defined binary operation (vector addition) on the set of equivalences classes.