units in math, cross product












1














The cross product of two vectors has length equal to the area of the parallelogram they generate.



The cross product is also a vector and thus has dimensions. But the units of those dimensions are units of area, such as m2.



My question is - this vector can't be in the same vector space as the original vectors multiplied, right?



What does adding units to dimension do to a vector space?










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  • Probably related: en.wikipedia.org/wiki/Pseudovector
    – lisyarus
    Sep 24 '15 at 9:44










  • ... or en.wikipedia.org/wiki/Bivector.
    – Hans Lundmark
    Sep 24 '15 at 9:59










  • A similar question: math.stackexchange.com/questions/705954/…
    – Hans Lundmark
    Sep 24 '15 at 10:01










  • Also seems helpful: mathoverflow.net/questions/4069/...
    – Scorpion_God
    Sep 24 '15 at 11:32










  • @JoonatanTalviste You might also find Muphrid's answer to this question informative.
    – user137731
    Oct 1 '15 at 16:59
















1














The cross product of two vectors has length equal to the area of the parallelogram they generate.



The cross product is also a vector and thus has dimensions. But the units of those dimensions are units of area, such as m2.



My question is - this vector can't be in the same vector space as the original vectors multiplied, right?



What does adding units to dimension do to a vector space?










share|cite|improve this question






















  • Probably related: en.wikipedia.org/wiki/Pseudovector
    – lisyarus
    Sep 24 '15 at 9:44










  • ... or en.wikipedia.org/wiki/Bivector.
    – Hans Lundmark
    Sep 24 '15 at 9:59










  • A similar question: math.stackexchange.com/questions/705954/…
    – Hans Lundmark
    Sep 24 '15 at 10:01










  • Also seems helpful: mathoverflow.net/questions/4069/...
    – Scorpion_God
    Sep 24 '15 at 11:32










  • @JoonatanTalviste You might also find Muphrid's answer to this question informative.
    – user137731
    Oct 1 '15 at 16:59














1












1








1


0





The cross product of two vectors has length equal to the area of the parallelogram they generate.



The cross product is also a vector and thus has dimensions. But the units of those dimensions are units of area, such as m2.



My question is - this vector can't be in the same vector space as the original vectors multiplied, right?



What does adding units to dimension do to a vector space?










share|cite|improve this question













The cross product of two vectors has length equal to the area of the parallelogram they generate.



The cross product is also a vector and thus has dimensions. But the units of those dimensions are units of area, such as m2.



My question is - this vector can't be in the same vector space as the original vectors multiplied, right?



What does adding units to dimension do to a vector space?







cross-product






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Sep 24 '15 at 9:34









Joonatan TalvisteJoonatan Talviste

61




61












  • Probably related: en.wikipedia.org/wiki/Pseudovector
    – lisyarus
    Sep 24 '15 at 9:44










  • ... or en.wikipedia.org/wiki/Bivector.
    – Hans Lundmark
    Sep 24 '15 at 9:59










  • A similar question: math.stackexchange.com/questions/705954/…
    – Hans Lundmark
    Sep 24 '15 at 10:01










  • Also seems helpful: mathoverflow.net/questions/4069/...
    – Scorpion_God
    Sep 24 '15 at 11:32










  • @JoonatanTalviste You might also find Muphrid's answer to this question informative.
    – user137731
    Oct 1 '15 at 16:59


















  • Probably related: en.wikipedia.org/wiki/Pseudovector
    – lisyarus
    Sep 24 '15 at 9:44










  • ... or en.wikipedia.org/wiki/Bivector.
    – Hans Lundmark
    Sep 24 '15 at 9:59










  • A similar question: math.stackexchange.com/questions/705954/…
    – Hans Lundmark
    Sep 24 '15 at 10:01










  • Also seems helpful: mathoverflow.net/questions/4069/...
    – Scorpion_God
    Sep 24 '15 at 11:32










  • @JoonatanTalviste You might also find Muphrid's answer to this question informative.
    – user137731
    Oct 1 '15 at 16:59
















Probably related: en.wikipedia.org/wiki/Pseudovector
– lisyarus
Sep 24 '15 at 9:44




Probably related: en.wikipedia.org/wiki/Pseudovector
– lisyarus
Sep 24 '15 at 9:44












... or en.wikipedia.org/wiki/Bivector.
– Hans Lundmark
Sep 24 '15 at 9:59




... or en.wikipedia.org/wiki/Bivector.
– Hans Lundmark
Sep 24 '15 at 9:59












A similar question: math.stackexchange.com/questions/705954/…
– Hans Lundmark
Sep 24 '15 at 10:01




A similar question: math.stackexchange.com/questions/705954/…
– Hans Lundmark
Sep 24 '15 at 10:01












Also seems helpful: mathoverflow.net/questions/4069/...
– Scorpion_God
Sep 24 '15 at 11:32




Also seems helpful: mathoverflow.net/questions/4069/...
– Scorpion_God
Sep 24 '15 at 11:32












@JoonatanTalviste You might also find Muphrid's answer to this question informative.
– user137731
Oct 1 '15 at 16:59




@JoonatanTalviste You might also find Muphrid's answer to this question informative.
– user137731
Oct 1 '15 at 16:59










1 Answer
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Yes -- using a vector to write the cross product is just a 3D convention to remind you it has 3 independent components. It's more naturally represented as a bivector or as a rank-2 tensor, but there's a duality between these and vectors in 3D.



Scale invariance/tensors in physics



The formal way of expressing your concern -- that the cross product has different "units" from other vectors in your space -- is that the cross product doesn't behave like a vector under scaling. Under scaling, a vector is supposed to scale as $vtolambda v$, because lengths scale that way, and areas scale as $ttolambda^2t$. The cross product scales as an area, so it makes a sucky vector.



This is sort of related to the definition of tensors in physics (specifically in relativity) -- a tensor is an object that transforms as a tensor under specific transformations, specifically Lorentz transformations (skews in the t-x/t-y/t-z plane and rotations in the other three, x-y/y-z/z-x). So scalars are invariant under Lorentz transformations, vectors transform as $Lambda_mu^{barmu}v^mu$, rank-2 tensors transform as $Lambda_mu^{barmu}Lambda_nu^{barnu}t^{munu}$, etc. So it's not enough to have the right number of components -- because this can be gamed with symmetries, like it is for the cross product (which has nine components, but only three independent one) -- you need to transform in the right way.



It's a bit more complicated with scaling, because different scalars transform differently (lengths scale like vectors, areas scale like rank-2 tensors), but the idea is the same -- the cross product is not a vector in the physics sense, although it is in the math sense. But nobody in math cares about the cross product anyway.






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    Yes -- using a vector to write the cross product is just a 3D convention to remind you it has 3 independent components. It's more naturally represented as a bivector or as a rank-2 tensor, but there's a duality between these and vectors in 3D.



    Scale invariance/tensors in physics



    The formal way of expressing your concern -- that the cross product has different "units" from other vectors in your space -- is that the cross product doesn't behave like a vector under scaling. Under scaling, a vector is supposed to scale as $vtolambda v$, because lengths scale that way, and areas scale as $ttolambda^2t$. The cross product scales as an area, so it makes a sucky vector.



    This is sort of related to the definition of tensors in physics (specifically in relativity) -- a tensor is an object that transforms as a tensor under specific transformations, specifically Lorentz transformations (skews in the t-x/t-y/t-z plane and rotations in the other three, x-y/y-z/z-x). So scalars are invariant under Lorentz transformations, vectors transform as $Lambda_mu^{barmu}v^mu$, rank-2 tensors transform as $Lambda_mu^{barmu}Lambda_nu^{barnu}t^{munu}$, etc. So it's not enough to have the right number of components -- because this can be gamed with symmetries, like it is for the cross product (which has nine components, but only three independent one) -- you need to transform in the right way.



    It's a bit more complicated with scaling, because different scalars transform differently (lengths scale like vectors, areas scale like rank-2 tensors), but the idea is the same -- the cross product is not a vector in the physics sense, although it is in the math sense. But nobody in math cares about the cross product anyway.






    share|cite|improve this answer




























      0














      Yes -- using a vector to write the cross product is just a 3D convention to remind you it has 3 independent components. It's more naturally represented as a bivector or as a rank-2 tensor, but there's a duality between these and vectors in 3D.



      Scale invariance/tensors in physics



      The formal way of expressing your concern -- that the cross product has different "units" from other vectors in your space -- is that the cross product doesn't behave like a vector under scaling. Under scaling, a vector is supposed to scale as $vtolambda v$, because lengths scale that way, and areas scale as $ttolambda^2t$. The cross product scales as an area, so it makes a sucky vector.



      This is sort of related to the definition of tensors in physics (specifically in relativity) -- a tensor is an object that transforms as a tensor under specific transformations, specifically Lorentz transformations (skews in the t-x/t-y/t-z plane and rotations in the other three, x-y/y-z/z-x). So scalars are invariant under Lorentz transformations, vectors transform as $Lambda_mu^{barmu}v^mu$, rank-2 tensors transform as $Lambda_mu^{barmu}Lambda_nu^{barnu}t^{munu}$, etc. So it's not enough to have the right number of components -- because this can be gamed with symmetries, like it is for the cross product (which has nine components, but only three independent one) -- you need to transform in the right way.



      It's a bit more complicated with scaling, because different scalars transform differently (lengths scale like vectors, areas scale like rank-2 tensors), but the idea is the same -- the cross product is not a vector in the physics sense, although it is in the math sense. But nobody in math cares about the cross product anyway.






      share|cite|improve this answer


























        0












        0








        0






        Yes -- using a vector to write the cross product is just a 3D convention to remind you it has 3 independent components. It's more naturally represented as a bivector or as a rank-2 tensor, but there's a duality between these and vectors in 3D.



        Scale invariance/tensors in physics



        The formal way of expressing your concern -- that the cross product has different "units" from other vectors in your space -- is that the cross product doesn't behave like a vector under scaling. Under scaling, a vector is supposed to scale as $vtolambda v$, because lengths scale that way, and areas scale as $ttolambda^2t$. The cross product scales as an area, so it makes a sucky vector.



        This is sort of related to the definition of tensors in physics (specifically in relativity) -- a tensor is an object that transforms as a tensor under specific transformations, specifically Lorentz transformations (skews in the t-x/t-y/t-z plane and rotations in the other three, x-y/y-z/z-x). So scalars are invariant under Lorentz transformations, vectors transform as $Lambda_mu^{barmu}v^mu$, rank-2 tensors transform as $Lambda_mu^{barmu}Lambda_nu^{barnu}t^{munu}$, etc. So it's not enough to have the right number of components -- because this can be gamed with symmetries, like it is for the cross product (which has nine components, but only three independent one) -- you need to transform in the right way.



        It's a bit more complicated with scaling, because different scalars transform differently (lengths scale like vectors, areas scale like rank-2 tensors), but the idea is the same -- the cross product is not a vector in the physics sense, although it is in the math sense. But nobody in math cares about the cross product anyway.






        share|cite|improve this answer














        Yes -- using a vector to write the cross product is just a 3D convention to remind you it has 3 independent components. It's more naturally represented as a bivector or as a rank-2 tensor, but there's a duality between these and vectors in 3D.



        Scale invariance/tensors in physics



        The formal way of expressing your concern -- that the cross product has different "units" from other vectors in your space -- is that the cross product doesn't behave like a vector under scaling. Under scaling, a vector is supposed to scale as $vtolambda v$, because lengths scale that way, and areas scale as $ttolambda^2t$. The cross product scales as an area, so it makes a sucky vector.



        This is sort of related to the definition of tensors in physics (specifically in relativity) -- a tensor is an object that transforms as a tensor under specific transformations, specifically Lorentz transformations (skews in the t-x/t-y/t-z plane and rotations in the other three, x-y/y-z/z-x). So scalars are invariant under Lorentz transformations, vectors transform as $Lambda_mu^{barmu}v^mu$, rank-2 tensors transform as $Lambda_mu^{barmu}Lambda_nu^{barnu}t^{munu}$, etc. So it's not enough to have the right number of components -- because this can be gamed with symmetries, like it is for the cross product (which has nine components, but only three independent one) -- you need to transform in the right way.



        It's a bit more complicated with scaling, because different scalars transform differently (lengths scale like vectors, areas scale like rank-2 tensors), but the idea is the same -- the cross product is not a vector in the physics sense, although it is in the math sense. But nobody in math cares about the cross product anyway.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 30 '18 at 20:03

























        answered Jun 9 '18 at 15:36









        Abhimanyu Pallavi SudhirAbhimanyu Pallavi Sudhir

        864619




        864619






























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