Vectors and points; abstract differences [duplicate]
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Distinction between vectors and points
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I would ask this on stackoverflow but I wanted a more theoretical answer. I understand that vectors have magnitude and direction whereas points are just coordinates.
In computing and especially in data science we use vectors to represent objects with multiple attributes. But as I mentioned vectors have magnitude and directions and it seems to me that in order to describe the attributes or properties of an object it would be better if we used points.
Why do use vectors then?
abstract-algebra vectors data-analysis
asked Aug 15 at 12:27
Harry Touloupas
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marked as duplicate by rschwieb abstract-algebra
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Aug 15 at 13:21
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This question already has an answer here:
Distinction between vectors and points
5 answers
I would ask this on stackoverflow but I wanted a more theoretical answer. I understand that vectors have magnitude and direction whereas points are just coordinates.
In computing and especially in data science we use vectors to represent objects with multiple attributes. But as I mentioned vectors have magnitude and directions and it seems to me that in order to describe the attributes or properties of an object it would be better if we used points.
Why do use vectors then?
abstract-algebra vectors data-analysis
asked Aug 15 at 12:27
Harry Touloupas
104
marked as duplicate by rschwieb abstract-algebra
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Aug 15 at 13:21
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Nothing in the first paragraph is entirely right, but it is useful in certain contexts, like $mathbb R^n$. Misleading, though, if you take it as scripture. "vector" is what you call an element of a vector space. "point" is what you call an element of a set when you have some geometric or topological view of the set. "point" is more like a location, and "vector" is "like a direction" although for certain vector spaces direction is undefined.
– rschwieb
Aug 15 at 13:24
In vector algebra, we can represent points via position vectors for example. The vector essentially points out (no pun intended) the location with respect to the origin. That is why the terms are sometimes used interchangeably. Really, there is not a whole lot of utility from worrying about the difference. Just keep in mind that there are two terms and try to slowly develop the difference between the two in your mind.
– rschwieb
Aug 15 at 13:25
add a comment |Â
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up vote
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down vote
favorite
This question already has an answer here:
Distinction between vectors and points
5 answers
I would ask this on stackoverflow but I wanted a more theoretical answer. I understand that vectors have magnitude and direction whereas points are just coordinates.
In computing and especially in data science we use vectors to represent objects with multiple attributes. But as I mentioned vectors have magnitude and directions and it seems to me that in order to describe the attributes or properties of an object it would be better if we used points.
Why do use vectors then?
abstract-algebra vectors data-analysis
asked Aug 15 at 12:27
Harry Touloupas
104
This question already has an answer here:
Distinction between vectors and points
5 answers
I would ask this on stackoverflow but I wanted a more theoretical answer. I understand that vectors have magnitude and direction whereas points are just coordinates.
In computing and especially in data science we use vectors to represent objects with multiple attributes. But as I mentioned vectors have magnitude and directions and it seems to me that in order to describe the attributes or properties of an object it would be better if we used points.
Why do use vectors then?
This question already has an answer here:
Distinction between vectors and points
5 answers
abstract-algebra vectors data-analysis
asked Aug 15 at 12:27
Harry Touloupas
104
asked Aug 15 at 12:27
Harry Touloupas
104
asked Aug 15 at 12:27
Harry Touloupas
104
asked Aug 15 at 12:27
Harry Touloupas
104
104
marked as duplicate by rschwieb abstract-algebra
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Aug 15 at 13:21
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Nothing in the first paragraph is entirely right, but it is useful in certain contexts, like $mathbb R^n$. Misleading, though, if you take it as scripture. "vector" is what you call an element of a vector space. "point" is what you call an element of a set when you have some geometric or topological view of the set. "point" is more like a location, and "vector" is "like a direction" although for certain vector spaces direction is undefined.
– rschwieb
Aug 15 at 13:24
In vector algebra, we can represent points via position vectors for example. The vector essentially points out (no pun intended) the location with respect to the origin. That is why the terms are sometimes used interchangeably. Really, there is not a whole lot of utility from worrying about the difference. Just keep in mind that there are two terms and try to slowly develop the difference between the two in your mind.
– rschwieb
Aug 15 at 13:25
add a comment |Â
Nothing in the first paragraph is entirely right, but it is useful in certain contexts, like $mathbb R^n$. Misleading, though, if you take it as scripture. "vector" is what you call an element of a vector space. "point" is what you call an element of a set when you have some geometric or topological view of the set. "point" is more like a location, and "vector" is "like a direction" although for certain vector spaces direction is undefined.
– rschwieb
Aug 15 at 13:24
In vector algebra, we can represent points via position vectors for example. The vector essentially points out (no pun intended) the location with respect to the origin. That is why the terms are sometimes used interchangeably. Really, there is not a whole lot of utility from worrying about the difference. Just keep in mind that there are two terms and try to slowly develop the difference between the two in your mind.
– rschwieb
Aug 15 at 13:25
Nothing in the first paragraph is entirely right, but it is useful in certain contexts, like $mathbb R^n$. Misleading, though, if you take it as scripture. "vector" is what you call an element of a vector space. "point" is what you call an element of a set when you have some geometric or topological view of the set. "point" is more like a location, and "vector" is "like a direction" although for certain vector spaces direction is undefined.
– rschwieb
Aug 15 at 13:24
Nothing in the first paragraph is entirely right, but it is useful in certain contexts, like $mathbb R^n$. Misleading, though, if you take it as scripture. "vector" is what you call an element of a vector space. "point" is what you call an element of a set when you have some geometric or topological view of the set. "point" is more like a location, and "vector" is "like a direction" although for certain vector spaces direction is undefined.
– rschwieb
Aug 15 at 13:24
In vector algebra, we can represent points via position vectors for example. The vector essentially points out (no pun intended) the location with respect to the origin. That is why the terms are sometimes used interchangeably. Really, there is not a whole lot of utility from worrying about the difference. Just keep in mind that there are two terms and try to slowly develop the difference between the two in your mind.
– rschwieb
Aug 15 at 13:25
In vector algebra, we can represent points via position vectors for example. The vector essentially points out (no pun intended) the location with respect to the origin. That is why the terms are sometimes used interchangeably. Really, there is not a whole lot of utility from worrying about the difference. Just keep in mind that there are two terms and try to slowly develop the difference between the two in your mind.
– rschwieb
Aug 15 at 13:25
add a comment |Â
1 Answer
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There is no "theoretical answer" to this question.
Mathematically speaking, each of the abstract objects you call "points" or "vectors" can be described by a list of numbers once you have chosen a coordinate system.
In data science the coordinate system is usually built into the problem: one coordinate for each numerical attribute. That you sometimes call the resulting list of numbers a "vector" does not mean you want to think of it as having length and direction.
Sometimes vector operations on data points may be useful. I think that's the case when analyzing some high dimensional data. Then the dot product between vectors/points can tell you the angle between them, which tells you when they are roughly proportional or nearly perpendicular. That can have semantic consequences in your data analysis. (Note: this is an impression I have from talking with folks who do this kind of work. It's not a definitive assertion.)
(although sometimes that is in fact useful).
answered Aug 15 at 12:45
Ethan Bolker
36k54299
Example where it is useful?
– Harry Touloupas
Aug 15 at 13:02
Please see my edit .
– Ethan Bolker
Aug 15 at 13:21
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
There is no "theoretical answer" to this question.
Mathematically speaking, each of the abstract objects you call "points" or "vectors" can be described by a list of numbers once you have chosen a coordinate system.
In data science the coordinate system is usually built into the problem: one coordinate for each numerical attribute. That you sometimes call the resulting list of numbers a "vector" does not mean you want to think of it as having length and direction.
Sometimes vector operations on data points may be useful. I think that's the case when analyzing some high dimensional data. Then the dot product between vectors/points can tell you the angle between them, which tells you when they are roughly proportional or nearly perpendicular. That can have semantic consequences in your data analysis. (Note: this is an impression I have from talking with folks who do this kind of work. It's not a definitive assertion.)
(although sometimes that is in fact useful).
answered Aug 15 at 12:45
Ethan Bolker
36k54299
Example where it is useful?
– Harry Touloupas
Aug 15 at 13:02
Please see my edit .
– Ethan Bolker
Aug 15 at 13:21
add a comment |Â
up vote
0
down vote
There is no "theoretical answer" to this question.
Mathematically speaking, each of the abstract objects you call "points" or "vectors" can be described by a list of numbers once you have chosen a coordinate system.
In data science the coordinate system is usually built into the problem: one coordinate for each numerical attribute. That you sometimes call the resulting list of numbers a "vector" does not mean you want to think of it as having length and direction.
Sometimes vector operations on data points may be useful. I think that's the case when analyzing some high dimensional data. Then the dot product between vectors/points can tell you the angle between them, which tells you when they are roughly proportional or nearly perpendicular. That can have semantic consequences in your data analysis. (Note: this is an impression I have from talking with folks who do this kind of work. It's not a definitive assertion.)
(although sometimes that is in fact useful).
answered Aug 15 at 12:45
Ethan Bolker
36k54299
Example where it is useful?
– Harry Touloupas
Aug 15 at 13:02
Please see my edit .
– Ethan Bolker
Aug 15 at 13:21
add a comment |Â
up vote
0
down vote
up vote
0
down vote
There is no "theoretical answer" to this question.
Mathematically speaking, each of the abstract objects you call "points" or "vectors" can be described by a list of numbers once you have chosen a coordinate system.
In data science the coordinate system is usually built into the problem: one coordinate for each numerical attribute. That you sometimes call the resulting list of numbers a "vector" does not mean you want to think of it as having length and direction.
Sometimes vector operations on data points may be useful. I think that's the case when analyzing some high dimensional data. Then the dot product between vectors/points can tell you the angle between them, which tells you when they are roughly proportional or nearly perpendicular. That can have semantic consequences in your data analysis. (Note: this is an impression I have from talking with folks who do this kind of work. It's not a definitive assertion.)
(although sometimes that is in fact useful).
answered Aug 15 at 12:45
Ethan Bolker
36k54299
There is no "theoretical answer" to this question.
Mathematically speaking, each of the abstract objects you call "points" or "vectors" can be described by a list of numbers once you have chosen a coordinate system.
In data science the coordinate system is usually built into the problem: one coordinate for each numerical attribute. That you sometimes call the resulting list of numbers a "vector" does not mean you want to think of it as having length and direction.
Sometimes vector operations on data points may be useful. I think that's the case when analyzing some high dimensional data. Then the dot product between vectors/points can tell you the angle between them, which tells you when they are roughly proportional or nearly perpendicular. That can have semantic consequences in your data analysis. (Note: this is an impression I have from talking with folks who do this kind of work. It's not a definitive assertion.)
(although sometimes that is in fact useful).
answered Aug 15 at 12:45
Ethan Bolker
36k54299
edited Aug 15 at 13:20
answered Aug 15 at 12:45
Ethan Bolker
36k54299
answered Aug 15 at 12:45
Ethan Bolker
36k54299
answered Aug 15 at 12:45
Ethan Bolker
36k54299
36k54299
Example where it is useful?
– Harry Touloupas
Aug 15 at 13:02
Please see my edit .
– Ethan Bolker
Aug 15 at 13:21
add a comment |Â
Example where it is useful?
– Harry Touloupas
Aug 15 at 13:02
Please see my edit .
– Ethan Bolker
Aug 15 at 13:21
Example where it is useful?
– Harry Touloupas
Aug 15 at 13:02
Example where it is useful?
– Harry Touloupas
Aug 15 at 13:02
Please see my edit .
– Ethan Bolker
Aug 15 at 13:21
Please see my edit .
– Ethan Bolker
Aug 15 at 13:21
add a comment |Â
Nothing in the first paragraph is entirely right, but it is useful in certain contexts, like $mathbb R^n$. Misleading, though, if you take it as scripture. "vector" is what you call an element of a vector space. "point" is what you call an element of a set when you have some geometric or topological view of the set. "point" is more like a location, and "vector" is "like a direction" although for certain vector spaces direction is undefined.
– rschwieb
Aug 15 at 13:24
In vector algebra, we can represent points via position vectors for example. The vector essentially points out (no pun intended) the location with respect to the origin. That is why the terms are sometimes used interchangeably. Really, there is not a whole lot of utility from worrying about the difference. Just keep in mind that there are two terms and try to slowly develop the difference between the two in your mind.
– rschwieb
Aug 15 at 13:25