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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.

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?

tryst with freedom
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  • You mean the origin as in "Where do vectors come from and what motivated them", or the actual origin of the 3D Cartesian space? – Eduardo Magalhães Jul 10 '20 at 19:00
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    The actual origin of 3-d space, i.e where the vectors tails are attached too – tryst with freedom Jul 10 '20 at 19:00
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    Please take some time to clarify your specific question. "Why can we do this and why does eveyrthing still work out after we do this?" is pretty squishy. – rschwieb Jul 10 '20 at 19:06
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    Well. What you learned in school was correct... when you talk about vectors in math. There are, however, certain fields of physics and similar where you essentially define a vector as the mathematical vector plus a point describing the origin of the vector. It's really just two different meanings of the same word. Everything works out so fine because in the important parts, one secretly uses the math definition in physics, too. – NeitherNor Jul 10 '20 at 19:07
  • Why .... shouldn't it work. .... Here's an analogy. If I get in my car and start driving and at the moment I pass exit #314 I am traveling 55.6 mph.. My speed is 55.6 mph. Meanwhile someone else on the other side of the world is driving a car and the moment they pass ausfahrt #QA7 they are traveling 55.6 mph too. But how can we have the same speed if we are in two different places? Where do these speeds exist? Are the just floating in space? How do we attach them to cars occaionally. Your question isn't bad but... figure out exactly what you are asking. That's half the job. – fleablood Jul 10 '20 at 19:13
  • Ok yeah you described the problem nicely, but my problem is describing it like more precisely – tryst with freedom Jul 10 '20 at 19:18
  • This might help: https://math.stackexchange.com/questions/645672/what-is-the-difference-between-a-point-and-a-vector – bubba Aug 21 '20 at 11:49
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    If you are going to study advanced mathematics and physics, be aware that the definitive answer is in differential geometry: every point has attached a vector space, and vectors pinned to different points cannot be added because they belong in different universes, so to speak. – Miguel Mar 19 '22 at 14:06
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    Encountered that already xD. I found the book Visual Differential GEometry by Tristan Needham to be a gentle introduction @Miguel – tryst with freedom Mar 19 '22 at 14:08
  • My answer is here. – ryang Mar 20 '22 at 11:10

4 Answers4

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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.

DreiCleaner
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  • Pleas use the angular brackets \langle $\langle$ and \rangle $\rangle$ instead of the comparison operators $<$ and $>$. – Christoph Aug 21 '20 at 12:32
  • I think this is a pretty myopic way to see things. This is a possible way to work with vectors for sure, but it is by no means the only sense in which the term is used. We often also want to work with base-pointed vectors --- for which we come up with the idea of tangent vectors to a smooth manifold, etc. – YiFan Tey Mar 19 '22 at 14:01
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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.)

littleO
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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.

Eduardo Magalhães
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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.

asd.123
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  • what are these conditions that you speak of – tryst with freedom Jul 11 '20 at 08:45
  • @DDD4C4U the conditions can be generalized (for mechanics) as, if the force is transmitted through a body that has a rigid structure, will transmit the force itself without changing its direction and magnitude but the position you apply the force and the point that you dot product two vectors can be different – asd.123 Jul 11 '20 at 08:56
  • If you want to grasp the whole idea of non-pseudo vectors that are used in physics I recommend you to check the Feynman's lectures volume 1 for the mechanics. – asd.123 Jul 11 '20 at 08:58
  • I'm just trying to say that mathematical operations done in physics especially in mechanics have quite logical explanations but they uses the physical properties of well defined materials so if you your question specially dedicated for this idea then you also need to analyse the structures that uses the mathematics. – asd.123 Jul 11 '20 at 09:04
  • you need to analyse the structure that uses mathematics... what does this mean? and what chapter of feynman shuld I read

    Also do you have suggestions for improving my question coz I feel like u understood it

     – tryst with freedom Jul 11 '20 at 16:19