Why do sometimes care for where vectors originate from and sometimes not? and exactly how many kinds of vectors are there?

Question

When I did linear algebra in high-school, it wasn't of much importance where the vectors originated from and for me this is a really hard concept to grasp. It's like no matter where the two vectors are pivoted in 3-d space, their dot product is invariant.

Like, we don't even define an origin when talking about vectors... it's like they're freely floating in space. Why can we do this as in why do we not need to regard origin when we speak of vectors?

Is the vector attached to some object? like does it not matter where the 'tail' is.

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Some more context

This question arose mainly when I was learning about plotting vector fields, in that, I had to associate each point with a vector so definitely here the vectors origin is relevant but not in the previous case, why?

I had also come across this problem when studying physics, see this post. The person answering the post says that the cross product gives an axial vector. So I wonder how many types of vectors are there?

Does this mean that regular 'vector' that we learned of has many 'cousin-forms'? How many takes types of vectors are there? how do we distinguish between these kinds of vectors?

Answer 1

So, vector are free to move in space.

Vectors are very different than points: With points we do care about their position in the plane/space.

If we have the point $A = (1,2)$ and $B = (3,2)$ then $A \neq B$.

But when we use vectors to study something, we usually just want a scene of direction, thus it does no matter where the tail is, it only matters where it points and his magnitude (length).

If $\bar A$ is the vector from $(1,1)$ to $(2,2)$ (or $\bar A = (1,1)$), and if $\bar B$ is the vector that points from $(3,3)$ to $(4,4)$, (or $\bar B = (1,1)$) then $\bar A = \bar B$, because with vectors we only care about the direction and the magnitude (length of the vector).

Depending on what you're trying to study you need to chose what mathematical tool is the best to assist you, and if you only care about things such as direction and not much about the specific position, vectors is the way to go.

Answer 2

In linear algebra all the the operations are well defined such as dot product. The reason that even though the vectors seems to be a floating objects (generally treated in introductory physics and as far as I understood from the question you are asking from this perspective) but when it comes to the operations they are always using such invariant order and process. To explain it with dot product, assume you have two vectors and you want to dot product them then looking at the definition of dot product you get a process, even though two vectors are completely separated in a vector space, that combines the vectors results in what is expected but in physical perspective dot product of two vectors that are not intersecting from head or tail also have the same dot product as in mathematics perspective because in physics many forces existing in nature have the feature of transferring with the same direction through a rigid body in an appropriate system. Also in introductory physics these conditions are met because the complexity of converse situation.

Answer 3

Vectors are defined by their magnitude and direction, not by their starting and ending points.

For that reason, the vector that starts at $(2,1)$ and ends at $(5,1)$ is the same vector as one that starts at $(0,0)$ and ends at $(3,4)$. They are both $\langle3,4\rangle$ or $\binom 34$, depending on what notation you prefer or your book uses. They represent a displacement of $3$ units in the $x$ direction and $4$ units in the $y$ direction.

Their magnitude is $5$. You can use trigonometry if you like to figure out the angle they make with the $x$ axis.

So when a question asks for the angle between two vectors, I find it helpful to picture them both starting at the origin. Moving the tail of a vector to the origin does not change the vector, after all.

Answer 4

In my mind, I define an ordered triple to be a list of three real numbers $(x,y,z)$. There are two ways to visualize an ordered triple: the "point picture" and the "vector picture".

In the point picture, the triple $(x,y,z)$ is visualized by drawing the point in 3D space whose coordinates are $(x,y,z)$. So in this picture, an ordered triple specifies a location in space.

In the vector picture, to visualize $(x,y,z)$, you first select a point $P$ in 3D space, arbitrarily. Starting at $P$, you move a distance $x$ in the direction of the $x$-axis, and a distance $y$ in the direction of the $y$-axis, and a distance $z$ in the direction of the $z$-axis. The point where you end up is called $Q$. Then you draw an arrow from $P$ to $Q$. In this picture, an ordered triple specifies the displacement from one location to another in space. If you had chosen a different starting point $P$, then you would have drawn a different arrow, but that different arrow would at least have the same magnitude and direction as the first arrow, and it would be an equally valid way to visualize the ordered triple $(x,y,z)$.

When I want to suggest that someone visualize an ordered triple using the point picture, I call the ordered triple a "point". When I want to suggest that someone visualize an ordered triple using the vector picture, I call the ordered triple a "vector". Either way, in this way of looking at things, both points and vectors are truly just ordered triples of real numbers. The only difference is what we visualize when we think about them. (I'm not perfectly consistent about this terminology, but I usually try to be.)

(The vector picture also suggests new operations to perform on ordered triples that are not suggested by the point picture. For example, it does not make sense to add together locations in space, but it does make sense to add together displacements.)