# Is there any entity that possess information of magnitude,direction,starting and ending points?

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?

• I don't think the wedge product does what you want. The information you want is precisely the two endpoints $A,B$. Commented Dec 5, 2019 at 3:56
• @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? Commented Dec 5, 2019 at 8:29
• 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. Commented Dec 5, 2019 at 21:50
• @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. Commented Dec 6, 2019 at 1:52
• 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. Commented Dec 6, 2019 at 2:36

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