Is there an angle between vectors in n > 3 dimensions?

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I can see why the dot product gives the angle between two vectors on $mathbb R^2$, and that the angle between two vectors on $mathbb R^3$ make sense, because you can take the plane defined by those two vectors, so it kind of falls back to $mathbb R^2$, but what about $mathbb R^1$ or $mathbb R^n$ for $n geq 4$? Is there such a thing as an angle in those other dimensions? I know there's such a thing as a distance, since by definition it's the length of a vector, but I can't grasp the concept of an angle then.




vectors




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  • 3




    In a word, yes.
    – Lord Shark the Unknown
    Jul 29 at 7:56






  • 1




    Maybe there are no angles in $n$ dimensions, but the dot product is still the product of the norms and the cosine of the angle ;-)
    – Yves Daoust
    Jul 29 at 8:57















up vote
2
down vote

favorite












I can see why the dot product gives the angle between two vectors on $mathbb R^2$, and that the angle between two vectors on $mathbb R^3$ make sense, because you can take the plane defined by those two vectors, so it kind of falls back to $mathbb R^2$, but what about $mathbb R^1$ or $mathbb R^n$ for $n geq 4$? Is there such a thing as an angle in those other dimensions? I know there's such a thing as a distance, since by definition it's the length of a vector, but I can't grasp the concept of an angle then.




vectors




share|cite|improve this question

















  • 3




    In a word, yes.
    – Lord Shark the Unknown
    Jul 29 at 7:56






  • 1




    Maybe there are no angles in $n$ dimensions, but the dot product is still the product of the norms and the cosine of the angle ;-)
    – Yves Daoust
    Jul 29 at 8:57













up vote
2
down vote

favorite









up vote
2
down vote

favorite











I can see why the dot product gives the angle between two vectors on $mathbb R^2$, and that the angle between two vectors on $mathbb R^3$ make sense, because you can take the plane defined by those two vectors, so it kind of falls back to $mathbb R^2$, but what about $mathbb R^1$ or $mathbb R^n$ for $n geq 4$? Is there such a thing as an angle in those other dimensions? I know there's such a thing as a distance, since by definition it's the length of a vector, but I can't grasp the concept of an angle then.




vectors




share|cite|improve this question













I can see why the dot product gives the angle between two vectors on $mathbb R^2$, and that the angle between two vectors on $mathbb R^3$ make sense, because you can take the plane defined by those two vectors, so it kind of falls back to $mathbb R^2$, but what about $mathbb R^1$ or $mathbb R^n$ for $n geq 4$? Is there such a thing as an angle in those other dimensions? I know there's such a thing as a distance, since by definition it's the length of a vector, but I can't grasp the concept of an angle then.





vectors





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share|cite|improve this question








edited Jul 29 at 8:32
























asked Jul 29 at 7:55









Luciano Santos

305




305







  • 3




    In a word, yes.
    – Lord Shark the Unknown
    Jul 29 at 7:56






  • 1




    Maybe there are no angles in $n$ dimensions, but the dot product is still the product of the norms and the cosine of the angle ;-)
    – Yves Daoust
    Jul 29 at 8:57













  • 3




    In a word, yes.
    – Lord Shark the Unknown
    Jul 29 at 7:56






  • 1




    Maybe there are no angles in $n$ dimensions, but the dot product is still the product of the norms and the cosine of the angle ;-)
    – Yves Daoust
    Jul 29 at 8:57








3




3




In a word, yes.
– Lord Shark the Unknown
Jul 29 at 7:56




In a word, yes.
– Lord Shark the Unknown
Jul 29 at 7:56




1




1




Maybe there are no angles in $n$ dimensions, but the dot product is still the product of the norms and the cosine of the angle ;-)
– Yves Daoust
Jul 29 at 8:57





Maybe there are no angles in $n$ dimensions, but the dot product is still the product of the norms and the cosine of the angle ;-)
– Yves Daoust
Jul 29 at 8:57











2 Answers
2






active

oldest

votes

















up vote
7
down vote



accepted










In any dimension, any two vectors which are not collinear, span a plane. The angle between them is then exactly the same as the angle between two vectors in the plane.






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  • Is that plane always on the "second dimension", i.e., informally can we project those two vectors is 2D plane?
    – Luciano Santos
    Jul 29 at 11:15










  • Yes, we can project these two vectors onto a two dimensional plane.
    – uniquesolution
    Jul 29 at 11:55






  • 4




    @LucianoSantos: It's not just that you can project them onto a 2D plane. You can project anything onto anything you want. What is important here is that they lie in a two-dimensional plane.
    – tomasz
    Jul 29 at 13:30

















up vote
2
down vote













The notion of an angle exists in a general inner product space for example (beyond $mathbb R^n$ and the dot product). In the case of dot product on two or three tuples, the angle concept coincides with the 'geometric' concept that we are first familier with, from school for example.



The same is true about the notion of perpendicular or orthogonal. The orthogonal then is not necessarilly the same as perpendicular in the usual geometric sense.






share|cite|improve this answer





















  • What does it measure, though? I mean, in a more abstract way. Some relationship between the vectors?
    – Luciano Santos
    Jul 29 at 8:33







  • 2




    @Luciano Santos The concept of orthogonal is important in general. As in the basic case ('usual' perpendicular) orthogonal vectors are still useful when we try to mininise certain vector lengths. Again we can project orthogonaly to minimise the 'distance' bwn two vectors. Here I may be working with two ortogonal matrices, so I don't try to 'understand' the geometric 'meaning' but appreciate the algebraic result.
    – AnyAD
    Jul 29 at 8:39











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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
7
down vote



accepted










In any dimension, any two vectors which are not collinear, span a plane. The angle between them is then exactly the same as the angle between two vectors in the plane.






share|cite|improve this answer





















  • Is that plane always on the "second dimension", i.e., informally can we project those two vectors is 2D plane?
    – Luciano Santos
    Jul 29 at 11:15










  • Yes, we can project these two vectors onto a two dimensional plane.
    – uniquesolution
    Jul 29 at 11:55






  • 4




    @LucianoSantos: It's not just that you can project them onto a 2D plane. You can project anything onto anything you want. What is important here is that they lie in a two-dimensional plane.
    – tomasz
    Jul 29 at 13:30














up vote
7
down vote



accepted










In any dimension, any two vectors which are not collinear, span a plane. The angle between them is then exactly the same as the angle between two vectors in the plane.






share|cite|improve this answer





















  • Is that plane always on the "second dimension", i.e., informally can we project those two vectors is 2D plane?
    – Luciano Santos
    Jul 29 at 11:15










  • Yes, we can project these two vectors onto a two dimensional plane.
    – uniquesolution
    Jul 29 at 11:55






  • 4




    @LucianoSantos: It's not just that you can project them onto a 2D plane. You can project anything onto anything you want. What is important here is that they lie in a two-dimensional plane.
    – tomasz
    Jul 29 at 13:30












up vote
7
down vote



accepted







up vote
7
down vote



accepted






In any dimension, any two vectors which are not collinear, span a plane. The angle between them is then exactly the same as the angle between two vectors in the plane.






share|cite|improve this answer













In any dimension, any two vectors which are not collinear, span a plane. The angle between them is then exactly the same as the angle between two vectors in the plane.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 29 at 8:00









uniquesolution

7,536721




7,536721











  • Is that plane always on the "second dimension", i.e., informally can we project those two vectors is 2D plane?
    – Luciano Santos
    Jul 29 at 11:15










  • Yes, we can project these two vectors onto a two dimensional plane.
    – uniquesolution
    Jul 29 at 11:55






  • 4




    @LucianoSantos: It's not just that you can project them onto a 2D plane. You can project anything onto anything you want. What is important here is that they lie in a two-dimensional plane.
    – tomasz
    Jul 29 at 13:30
















  • Is that plane always on the "second dimension", i.e., informally can we project those two vectors is 2D plane?
    – Luciano Santos
    Jul 29 at 11:15










  • Yes, we can project these two vectors onto a two dimensional plane.
    – uniquesolution
    Jul 29 at 11:55






  • 4




    @LucianoSantos: It's not just that you can project them onto a 2D plane. You can project anything onto anything you want. What is important here is that they lie in a two-dimensional plane.
    – tomasz
    Jul 29 at 13:30















Is that plane always on the "second dimension", i.e., informally can we project those two vectors is 2D plane?
– Luciano Santos
Jul 29 at 11:15




Is that plane always on the "second dimension", i.e., informally can we project those two vectors is 2D plane?
– Luciano Santos
Jul 29 at 11:15












Yes, we can project these two vectors onto a two dimensional plane.
– uniquesolution
Jul 29 at 11:55




Yes, we can project these two vectors onto a two dimensional plane.
– uniquesolution
Jul 29 at 11:55




4




4




@LucianoSantos: It's not just that you can project them onto a 2D plane. You can project anything onto anything you want. What is important here is that they lie in a two-dimensional plane.
– tomasz
Jul 29 at 13:30




@LucianoSantos: It's not just that you can project them onto a 2D plane. You can project anything onto anything you want. What is important here is that they lie in a two-dimensional plane.
– tomasz
Jul 29 at 13:30










up vote
2
down vote













The notion of an angle exists in a general inner product space for example (beyond $mathbb R^n$ and the dot product). In the case of dot product on two or three tuples, the angle concept coincides with the 'geometric' concept that we are first familier with, from school for example.



The same is true about the notion of perpendicular or orthogonal. The orthogonal then is not necessarilly the same as perpendicular in the usual geometric sense.






share|cite|improve this answer





















  • What does it measure, though? I mean, in a more abstract way. Some relationship between the vectors?
    – Luciano Santos
    Jul 29 at 8:33







  • 2




    @Luciano Santos The concept of orthogonal is important in general. As in the basic case ('usual' perpendicular) orthogonal vectors are still useful when we try to mininise certain vector lengths. Again we can project orthogonaly to minimise the 'distance' bwn two vectors. Here I may be working with two ortogonal matrices, so I don't try to 'understand' the geometric 'meaning' but appreciate the algebraic result.
    – AnyAD
    Jul 29 at 8:39















up vote
2
down vote













The notion of an angle exists in a general inner product space for example (beyond $mathbb R^n$ and the dot product). In the case of dot product on two or three tuples, the angle concept coincides with the 'geometric' concept that we are first familier with, from school for example.



The same is true about the notion of perpendicular or orthogonal. The orthogonal then is not necessarilly the same as perpendicular in the usual geometric sense.






share|cite|improve this answer





















  • What does it measure, though? I mean, in a more abstract way. Some relationship between the vectors?
    – Luciano Santos
    Jul 29 at 8:33







  • 2




    @Luciano Santos The concept of orthogonal is important in general. As in the basic case ('usual' perpendicular) orthogonal vectors are still useful when we try to mininise certain vector lengths. Again we can project orthogonaly to minimise the 'distance' bwn two vectors. Here I may be working with two ortogonal matrices, so I don't try to 'understand' the geometric 'meaning' but appreciate the algebraic result.
    – AnyAD
    Jul 29 at 8:39













up vote
2
down vote










up vote
2
down vote









The notion of an angle exists in a general inner product space for example (beyond $mathbb R^n$ and the dot product). In the case of dot product on two or three tuples, the angle concept coincides with the 'geometric' concept that we are first familier with, from school for example.



The same is true about the notion of perpendicular or orthogonal. The orthogonal then is not necessarilly the same as perpendicular in the usual geometric sense.






share|cite|improve this answer













The notion of an angle exists in a general inner product space for example (beyond $mathbb R^n$ and the dot product). In the case of dot product on two or three tuples, the angle concept coincides with the 'geometric' concept that we are first familier with, from school for example.



The same is true about the notion of perpendicular or orthogonal. The orthogonal then is not necessarilly the same as perpendicular in the usual geometric sense.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 29 at 8:09









AnyAD

1,451611




1,451611











  • What does it measure, though? I mean, in a more abstract way. Some relationship between the vectors?
    – Luciano Santos
    Jul 29 at 8:33







  • 2




    @Luciano Santos The concept of orthogonal is important in general. As in the basic case ('usual' perpendicular) orthogonal vectors are still useful when we try to mininise certain vector lengths. Again we can project orthogonaly to minimise the 'distance' bwn two vectors. Here I may be working with two ortogonal matrices, so I don't try to 'understand' the geometric 'meaning' but appreciate the algebraic result.
    – AnyAD
    Jul 29 at 8:39

















  • What does it measure, though? I mean, in a more abstract way. Some relationship between the vectors?
    – Luciano Santos
    Jul 29 at 8:33







  • 2




    @Luciano Santos The concept of orthogonal is important in general. As in the basic case ('usual' perpendicular) orthogonal vectors are still useful when we try to mininise certain vector lengths. Again we can project orthogonaly to minimise the 'distance' bwn two vectors. Here I may be working with two ortogonal matrices, so I don't try to 'understand' the geometric 'meaning' but appreciate the algebraic result.
    – AnyAD
    Jul 29 at 8:39
















What does it measure, though? I mean, in a more abstract way. Some relationship between the vectors?
– Luciano Santos
Jul 29 at 8:33





What does it measure, though? I mean, in a more abstract way. Some relationship between the vectors?
– Luciano Santos
Jul 29 at 8:33





2




2




@Luciano Santos The concept of orthogonal is important in general. As in the basic case ('usual' perpendicular) orthogonal vectors are still useful when we try to mininise certain vector lengths. Again we can project orthogonaly to minimise the 'distance' bwn two vectors. Here I may be working with two ortogonal matrices, so I don't try to 'understand' the geometric 'meaning' but appreciate the algebraic result.
– AnyAD
Jul 29 at 8:39





@Luciano Santos The concept of orthogonal is important in general. As in the basic case ('usual' perpendicular) orthogonal vectors are still useful when we try to mininise certain vector lengths. Again we can project orthogonaly to minimise the 'distance' bwn two vectors. Here I may be working with two ortogonal matrices, so I don't try to 'understand' the geometric 'meaning' but appreciate the algebraic result.
– AnyAD
Jul 29 at 8:39













 

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