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Is there any entity similar to vectors but also possess the starting and ending points ?

For instance, consider a plane $z = 4$, Suppose I want a vector starting from A$(0,0,4)$ and ending at B$(0,1,4)$ with origin O$(0,0,0)$ how do I represent this? / What entity possess this information?

Until now I was in a state that vector also contains the information of starting and ending points, and tried to calculate the vector $\vec{AB}$ by calculating $\vec{OB} - \vec{OA}$, the answer I got is $(0,1,0)$ which is parallel to the vector $\vec{AB}$ I thought of , it is then I recalled vectors only have magnitude and direction.

So is there any such entity?

Edit : I meant to ask a $3D$ entity so that if it exists, the algebra with this will be helpful and easier.

Edit : So, after a decent search, I encountered grassmann algebra, which states that if I take the wedge product of a point with a vector I can have bound vector, which is what I exactly wanted.

So, what I want to know now is, Is there anything similar to bound vector in 3D vector algebra?

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  • $\begingroup$ I don't think the wedge product does what you want. The information you want is precisely the two endpoints $A,B$. $\endgroup$
    – mr_e_man
    Commented Dec 5, 2019 at 3:56
  • $\begingroup$ @mr_e_man, Yes, you are right, but the wedge product of the point A and the vector parallel to AB gives me a bound vector between A and B so it must have the starting point, which means it contains point B indirectly, so isn't this what I wanted? $\endgroup$
    – John Paul
    Commented Dec 5, 2019 at 8:29
  • $\begingroup$ I don't know what you mean by a "bound vector". I know of several models of $n$-dimensional Euclidean space, using $n$ or $(n+1)$ or $(n+2)$-dimensional vector spaces. The wedge product of vectors has different interpretations, and carries different information, in each model. The wedge product in $(n+2)$-dimensional CGA may do what you want. $\endgroup$
    – mr_e_man
    Commented Dec 5, 2019 at 21:50
  • $\begingroup$ @mr_e_man, according to Wikipedia, "A vector with fixed initial and terminal point is called a bound vector". And by reading that I assumed the initial and terminal points must have to be something I need to specify beforehand just like what wedge product is doing with a point and a free vector. $\endgroup$
    – John Paul
    Commented Dec 6, 2019 at 1:52
  • $\begingroup$ The wedge product applies to vectors (or whatever you call things in abstract linear algebra, to avoid physical or geometric connotations). To take the wedge product with a point, you need to somehow represent the point as a vector. There are several ways of doing this. $\endgroup$
    – mr_e_man
    Commented Dec 6, 2019 at 2:36

1 Answer 1

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Actually there is.

The thing you are looking for stores information about starting and ending point of an applied vector, i.e. two points in $\mathbb R^3$. An ordered couple of points in $\mathbb R^3$ can be identified with a point in $\mathbb R^6$ in the following way:

$$A=(a_1,a_2,a_3);B=(b_1,b_2,b_3) \to (a_1,a_2,a_3,b_1,b_2,b_3)$$

So the thing you are looking for is actually a vector. Simply, in a bigger dimensional vector space.

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  • $\begingroup$ Thanks for this, but is there any entity in 3D itself? cause the algebra I might encounter with this 6D thing will be exhaustive I guess $\endgroup$
    – John Paul
    Commented Dec 4, 2019 at 23:45
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    $\begingroup$ It's not clear what you mean by an "entity in 3D". Any structure that meets your needs will have to be essentially six dimensional since it takes that many numbers to describe what you want. The best you can do is find a mathematical description that makes it easier to calculate or understand. The amount of algebra will be pretty much the same. $\endgroup$
    – Ethan Bolker
    Commented Dec 5, 2019 at 13:45
  • $\begingroup$ @EthanBolker , thanks for the info ! $\endgroup$
    – John Paul
    Commented Dec 6, 2019 at 1:46

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