Is the position of vector in space immaterial?

Question

Are length (magnitude) and direction the only features to quality an entity as a vector?

If I move a line segment($ls$) in space without changing its length and direction, then the resultant line segment $(ls')$ is also the same vector?

For example, the line segment joining $(0,0)- (3,4)$ and the line segment joining $(6,0)- (9,4)$ are same vectors or different vectors?

Answer 1

Yes, length and magnitude uniquely qualify a vector.
The vector from $(0,0)$ to $(3,4)$ and the vector from $(6,0)$ to $(9,4)$ have the same components: $(3, 4)$.

Since they make same angle with the axes and have the same magnitude, they are different representations of the same vector even though they are two different segments.

Vectors don't change under translation because the components don't change.

Answer 2

If you cared about the exact points where a line segment started and ended, then that would be "coordinate geometry".

With vectors, you don't really care about that - or rather, you don't need a fixed origin and a fixed destination to define a vector. All that's necessary is to specify a magnitude and a direction.

And the freedom that vectors give you allows you to make some problems in coordinate geometry very easy. For example, if I told you that $ABCD$ is a parallelogram with $A = (-1,2); \ B= (2,4); \ C = (3,7)$ and asked you to find the coordinates of $D$, that would be somewhat tedious with just the tools available to you in elementary coordinate geometry (determining equations of lines and their intersections, or working with lengths of line segments). But using vectors and the notion of a translation, it becomes an almost trivial problem, $\vec{CD} = \vec{BA} \implies \begin{pmatrix}x_D-3 \\y_D-7\end{pmatrix} = \begin{pmatrix}-3 \\-2\end{pmatrix} \implies D = (0,5)$

Answer 3

The given answers are exact without doubt.

But, let me place the "debate" on a different ground, that is IMHO worth of interest.

If you take into account the 3D line to which belong the points, say $A$ and $B$, you can consider the physical notion of "sliding vector", which mathematically speaking (rational mechanics) is called a "torsor" (http://math.ucr.edu/home/baez/torsors.html). What is it ? The couple

$$(\vec{AB},\vec{m}).\tag{1}$$

where $\vec{m}:=\vec{OA} \times \vec{AB}/\|\vec{AB}\|$ is a moment.

In a kind of reciprocal way, this type of couple (1) characterizes a single 3D oriented line (i.e., there is a unique oriented line with this vector and this moment). This is the basis of the important Plücker coordinates (https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates).

Answer 4

A vector, by definition, cannot have position. But note that we may use vectors to define positions -- but that's quite a different thing.

The prime example of vector is displacement, or step. That's actually where Hamilton coined the name from. A step carries only the information of how far it carries you, and in what sense. It doesn't matter whether the step carries you from home or from work, or wherever, so long as the length of the step and the sense in which it is directed are identical -- then it's just the same step. We only geometrically represent vectors by arrows, but note that if you want to fully visualise it, it is actually a field of parallel, equally long, and similarly directed arrows that's a vector. Thus, formally, these vectors are equivalence classes. But the intuitive idea is more kinematic than geometric, although both points of view are obviously related -- the idea is that of a step, a carrier, a vector. Steps in this sense are abstract, nonvisual. We only see their effects. Of course there are other realisation of vectors (as forces, velocities, position vectors describing points in space, etc., and even the fully abstract definition), but I only wanted you to see the origin in order to fully comprehend the idea.

Thus, if you have a band of soldiers in parade, all in step, it's the same sequence of vectors that describes the motion of the column, not different vectors for each soldier, no. A vector does not care about where it starts from. However, in geometry, it is sometimes convenient to fix a starting point (the origin) for our vectors. These are the ones called position vectors.

Finally, in coordinate space, a vector can be completely described by a tuple of dimensions, each specifying the step along each of the coordinate axes. But we can also describe points of such a space by tuples of numbers. Thus, we set up a correspondence between points in space and vectors in that space. In general, we soon begin to notice that many other objects that we didn't think of as vectors seem to possess the basic properties of the original vectors. By extension we call these vectors too, and it is one of the most fruitful generalisations in mathematics.