Characterising $langle x,yrangle = langle x,xrangle cdot langle y,yrangle$












1












$begingroup$



Is it possible to give a nice characterisation of all couples $(x,y) in mathbb{R}^{2n}$ such that :
$$langle x,yrangle = langle x,xranglelangle y,yrangle?$$




$langle,rangle$ is the euclidian inner product on $mathbb{R}^n$.



Here are some thoughts :




  • With the intuition : $langle x,yrangle = $ the length of the projection of $x$ onto $y times$ the length of $y$. It would mean that the above equality means :
    $$text{length projection of ... = the square of the length of }x times text{ the square of the length of }y$$
    But by C-S this is possible only if the length of $y$, or $x$ is less than $1$.


  • The condition reminds me the independence of random variables, but I don’t know if it helps.



Thank you !










share|cite|improve this question











$endgroup$












  • $begingroup$
    What do you want? For example when $n=1$ you want a "nice characterization" of numbers $x,y$ such that $xy=x^2y^2$. What answer would you want in this case?
    $endgroup$
    – GEdgar
    Dec 20 '18 at 12:23










  • $begingroup$
    @GEdgar $(x,0), (0,x), (x,1/x)$, it just mean solve.
    $endgroup$
    – Thinking
    Dec 20 '18 at 12:28










  • $begingroup$
    @Thinking I'm not sure describing the fibre over each $y$ is what the OP had in mind. As written, it is a quartic hypersurface in $mathbb{R}^{2n}$ and that's pretty much what we usually meant by characterisation.
    $endgroup$
    – user10354138
    Dec 20 '18 at 14:22


















1












$begingroup$



Is it possible to give a nice characterisation of all couples $(x,y) in mathbb{R}^{2n}$ such that :
$$langle x,yrangle = langle x,xranglelangle y,yrangle?$$




$langle,rangle$ is the euclidian inner product on $mathbb{R}^n$.



Here are some thoughts :




  • With the intuition : $langle x,yrangle = $ the length of the projection of $x$ onto $y times$ the length of $y$. It would mean that the above equality means :
    $$text{length projection of ... = the square of the length of }x times text{ the square of the length of }y$$
    But by C-S this is possible only if the length of $y$, or $x$ is less than $1$.


  • The condition reminds me the independence of random variables, but I don’t know if it helps.



Thank you !










share|cite|improve this question











$endgroup$












  • $begingroup$
    What do you want? For example when $n=1$ you want a "nice characterization" of numbers $x,y$ such that $xy=x^2y^2$. What answer would you want in this case?
    $endgroup$
    – GEdgar
    Dec 20 '18 at 12:23










  • $begingroup$
    @GEdgar $(x,0), (0,x), (x,1/x)$, it just mean solve.
    $endgroup$
    – Thinking
    Dec 20 '18 at 12:28










  • $begingroup$
    @Thinking I'm not sure describing the fibre over each $y$ is what the OP had in mind. As written, it is a quartic hypersurface in $mathbb{R}^{2n}$ and that's pretty much what we usually meant by characterisation.
    $endgroup$
    – user10354138
    Dec 20 '18 at 14:22
















1












1








1





$begingroup$



Is it possible to give a nice characterisation of all couples $(x,y) in mathbb{R}^{2n}$ such that :
$$langle x,yrangle = langle x,xranglelangle y,yrangle?$$




$langle,rangle$ is the euclidian inner product on $mathbb{R}^n$.



Here are some thoughts :




  • With the intuition : $langle x,yrangle = $ the length of the projection of $x$ onto $y times$ the length of $y$. It would mean that the above equality means :
    $$text{length projection of ... = the square of the length of }x times text{ the square of the length of }y$$
    But by C-S this is possible only if the length of $y$, or $x$ is less than $1$.


  • The condition reminds me the independence of random variables, but I don’t know if it helps.



Thank you !










share|cite|improve this question











$endgroup$





Is it possible to give a nice characterisation of all couples $(x,y) in mathbb{R}^{2n}$ such that :
$$langle x,yrangle = langle x,xranglelangle y,yrangle?$$




$langle,rangle$ is the euclidian inner product on $mathbb{R}^n$.



Here are some thoughts :




  • With the intuition : $langle x,yrangle = $ the length of the projection of $x$ onto $y times$ the length of $y$. It would mean that the above equality means :
    $$text{length projection of ... = the square of the length of }x times text{ the square of the length of }y$$
    But by C-S this is possible only if the length of $y$, or $x$ is less than $1$.


  • The condition reminds me the independence of random variables, but I don’t know if it helps.



Thank you !







linear-algebra probability abstract-algebra euclidean-geometry






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edited Dec 20 '18 at 12:11









user10354138

7,4322925




7,4322925










asked Dec 20 '18 at 11:42









JebfiffkkfnfolzbdJebfiffkkfnfolzbd

642




642












  • $begingroup$
    What do you want? For example when $n=1$ you want a "nice characterization" of numbers $x,y$ such that $xy=x^2y^2$. What answer would you want in this case?
    $endgroup$
    – GEdgar
    Dec 20 '18 at 12:23










  • $begingroup$
    @GEdgar $(x,0), (0,x), (x,1/x)$, it just mean solve.
    $endgroup$
    – Thinking
    Dec 20 '18 at 12:28










  • $begingroup$
    @Thinking I'm not sure describing the fibre over each $y$ is what the OP had in mind. As written, it is a quartic hypersurface in $mathbb{R}^{2n}$ and that's pretty much what we usually meant by characterisation.
    $endgroup$
    – user10354138
    Dec 20 '18 at 14:22




















  • $begingroup$
    What do you want? For example when $n=1$ you want a "nice characterization" of numbers $x,y$ such that $xy=x^2y^2$. What answer would you want in this case?
    $endgroup$
    – GEdgar
    Dec 20 '18 at 12:23










  • $begingroup$
    @GEdgar $(x,0), (0,x), (x,1/x)$, it just mean solve.
    $endgroup$
    – Thinking
    Dec 20 '18 at 12:28










  • $begingroup$
    @Thinking I'm not sure describing the fibre over each $y$ is what the OP had in mind. As written, it is a quartic hypersurface in $mathbb{R}^{2n}$ and that's pretty much what we usually meant by characterisation.
    $endgroup$
    – user10354138
    Dec 20 '18 at 14:22


















$begingroup$
What do you want? For example when $n=1$ you want a "nice characterization" of numbers $x,y$ such that $xy=x^2y^2$. What answer would you want in this case?
$endgroup$
– GEdgar
Dec 20 '18 at 12:23




$begingroup$
What do you want? For example when $n=1$ you want a "nice characterization" of numbers $x,y$ such that $xy=x^2y^2$. What answer would you want in this case?
$endgroup$
– GEdgar
Dec 20 '18 at 12:23












$begingroup$
@GEdgar $(x,0), (0,x), (x,1/x)$, it just mean solve.
$endgroup$
– Thinking
Dec 20 '18 at 12:28




$begingroup$
@GEdgar $(x,0), (0,x), (x,1/x)$, it just mean solve.
$endgroup$
– Thinking
Dec 20 '18 at 12:28












$begingroup$
@Thinking I'm not sure describing the fibre over each $y$ is what the OP had in mind. As written, it is a quartic hypersurface in $mathbb{R}^{2n}$ and that's pretty much what we usually meant by characterisation.
$endgroup$
– user10354138
Dec 20 '18 at 14:22






$begingroup$
@Thinking I'm not sure describing the fibre over each $y$ is what the OP had in mind. As written, it is a quartic hypersurface in $mathbb{R}^{2n}$ and that's pretty much what we usually meant by characterisation.
$endgroup$
– user10354138
Dec 20 '18 at 14:22












3 Answers
3






active

oldest

votes


















0












$begingroup$

There doesn't seem to be a nice charcarterization but you can do the following: write $x=cy+z$ $,,$ (1) with $z$ orthogonal to $y$. [We can always do this]. Using the fact that $|cy+z|^{2}=c^{2}|y|^{2}+|z|^{2}$ we can see from the given equation that $c$ satisfies a quadratic equation. Solving this gives $c$ and these steps are all reversible so whenever $x$ and $y$ have this form we get $langle x,yrangle =langle x,xrangle langle y,yrangle$. I think this is the best characterization we can get. [ We will be forced to impose a condition like $|z|leq |y|$ because a real number $c$ satisfying (1) surely exists and the discriminant of the quadratic has to be non-negative].






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    The equation $langle x,yrangle = langle x,xranglelangle y,yrangle$ is trivially satisfied if $x=0$ or $y=0$. Therefore let us suppose that $x ne 0 ne y$. If $ phi$ is the angle between $x$ and $y$, then we have



    $$ cos phi= frac{langle x,yrangle}{||x|| cdot ||y||}.$$



    Then it is easy to see that



    $$langle x,yrangle = langle x,xranglelangle y,yrangle iff langle x,yrangle = cos^2 phi.$$






    share|cite|improve this answer









    $endgroup$





















      0












      $begingroup$

      The pair $(x,y)$ satisfies $langle x,yrangle = langle x,xranglelangle y,yrangle$ when $y=0$. When $yneq 0$ we may set $hat y = y/langle y,yrangle$ to obtain the equation $langle x,hat y rangle = langle x,xrangle$ which is equivalent to $x$ and $hat y-x$ being orthogonal. This should remind you of Thales's theorem, as the set of all such $x$ is the sphere with diameter the line segment from $0$ to $hat y$. That is, it is the set of all $x$ satisfying
      $$
      left| x - frac{hat y}{2}right| = left| frac{hat y}{2}right|
      $$

      or in the original variables
      $$
      left| x - frac{y}{2langle y,yrangle}right| = left| frac{y}{2langle y,yrangle}right| = frac{1}{2|y|}.
      $$



      In summary for $y=0$ every $x$ is possible, for each $yneq 0$ you have all $x$ on the sphere with diameter the line segment from $0$ to $y/langle y,yrangle$.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        You really mean sphere not circle ($n$ is not necessarily $1$), and this is another way of phrasing it.
        $endgroup$
        – user10354138
        Dec 20 '18 at 14:19










      • $begingroup$
        You are right, thanks. I was thinking about $n=2$. Good thing Thales's theorem holds for arbitrary spheres ;-)
        $endgroup$
        – Christoph
        Dec 20 '18 at 15:46











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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      0












      $begingroup$

      There doesn't seem to be a nice charcarterization but you can do the following: write $x=cy+z$ $,,$ (1) with $z$ orthogonal to $y$. [We can always do this]. Using the fact that $|cy+z|^{2}=c^{2}|y|^{2}+|z|^{2}$ we can see from the given equation that $c$ satisfies a quadratic equation. Solving this gives $c$ and these steps are all reversible so whenever $x$ and $y$ have this form we get $langle x,yrangle =langle x,xrangle langle y,yrangle$. I think this is the best characterization we can get. [ We will be forced to impose a condition like $|z|leq |y|$ because a real number $c$ satisfying (1) surely exists and the discriminant of the quadratic has to be non-negative].






      share|cite|improve this answer









      $endgroup$


















        0












        $begingroup$

        There doesn't seem to be a nice charcarterization but you can do the following: write $x=cy+z$ $,,$ (1) with $z$ orthogonal to $y$. [We can always do this]. Using the fact that $|cy+z|^{2}=c^{2}|y|^{2}+|z|^{2}$ we can see from the given equation that $c$ satisfies a quadratic equation. Solving this gives $c$ and these steps are all reversible so whenever $x$ and $y$ have this form we get $langle x,yrangle =langle x,xrangle langle y,yrangle$. I think this is the best characterization we can get. [ We will be forced to impose a condition like $|z|leq |y|$ because a real number $c$ satisfying (1) surely exists and the discriminant of the quadratic has to be non-negative].






        share|cite|improve this answer









        $endgroup$
















          0












          0








          0





          $begingroup$

          There doesn't seem to be a nice charcarterization but you can do the following: write $x=cy+z$ $,,$ (1) with $z$ orthogonal to $y$. [We can always do this]. Using the fact that $|cy+z|^{2}=c^{2}|y|^{2}+|z|^{2}$ we can see from the given equation that $c$ satisfies a quadratic equation. Solving this gives $c$ and these steps are all reversible so whenever $x$ and $y$ have this form we get $langle x,yrangle =langle x,xrangle langle y,yrangle$. I think this is the best characterization we can get. [ We will be forced to impose a condition like $|z|leq |y|$ because a real number $c$ satisfying (1) surely exists and the discriminant of the quadratic has to be non-negative].






          share|cite|improve this answer









          $endgroup$



          There doesn't seem to be a nice charcarterization but you can do the following: write $x=cy+z$ $,,$ (1) with $z$ orthogonal to $y$. [We can always do this]. Using the fact that $|cy+z|^{2}=c^{2}|y|^{2}+|z|^{2}$ we can see from the given equation that $c$ satisfies a quadratic equation. Solving this gives $c$ and these steps are all reversible so whenever $x$ and $y$ have this form we get $langle x,yrangle =langle x,xrangle langle y,yrangle$. I think this is the best characterization we can get. [ We will be forced to impose a condition like $|z|leq |y|$ because a real number $c$ satisfying (1) surely exists and the discriminant of the quadratic has to be non-negative].







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 20 '18 at 12:23









          Kavi Rama MurthyKavi Rama Murthy

          60.6k42161




          60.6k42161























              0












              $begingroup$

              The equation $langle x,yrangle = langle x,xranglelangle y,yrangle$ is trivially satisfied if $x=0$ or $y=0$. Therefore let us suppose that $x ne 0 ne y$. If $ phi$ is the angle between $x$ and $y$, then we have



              $$ cos phi= frac{langle x,yrangle}{||x|| cdot ||y||}.$$



              Then it is easy to see that



              $$langle x,yrangle = langle x,xranglelangle y,yrangle iff langle x,yrangle = cos^2 phi.$$






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                The equation $langle x,yrangle = langle x,xranglelangle y,yrangle$ is trivially satisfied if $x=0$ or $y=0$. Therefore let us suppose that $x ne 0 ne y$. If $ phi$ is the angle between $x$ and $y$, then we have



                $$ cos phi= frac{langle x,yrangle}{||x|| cdot ||y||}.$$



                Then it is easy to see that



                $$langle x,yrangle = langle x,xranglelangle y,yrangle iff langle x,yrangle = cos^2 phi.$$






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  The equation $langle x,yrangle = langle x,xranglelangle y,yrangle$ is trivially satisfied if $x=0$ or $y=0$. Therefore let us suppose that $x ne 0 ne y$. If $ phi$ is the angle between $x$ and $y$, then we have



                  $$ cos phi= frac{langle x,yrangle}{||x|| cdot ||y||}.$$



                  Then it is easy to see that



                  $$langle x,yrangle = langle x,xranglelangle y,yrangle iff langle x,yrangle = cos^2 phi.$$






                  share|cite|improve this answer









                  $endgroup$



                  The equation $langle x,yrangle = langle x,xranglelangle y,yrangle$ is trivially satisfied if $x=0$ or $y=0$. Therefore let us suppose that $x ne 0 ne y$. If $ phi$ is the angle between $x$ and $y$, then we have



                  $$ cos phi= frac{langle x,yrangle}{||x|| cdot ||y||}.$$



                  Then it is easy to see that



                  $$langle x,yrangle = langle x,xranglelangle y,yrangle iff langle x,yrangle = cos^2 phi.$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 20 '18 at 12:42









                  FredFred

                  46.8k1848




                  46.8k1848























                      0












                      $begingroup$

                      The pair $(x,y)$ satisfies $langle x,yrangle = langle x,xranglelangle y,yrangle$ when $y=0$. When $yneq 0$ we may set $hat y = y/langle y,yrangle$ to obtain the equation $langle x,hat y rangle = langle x,xrangle$ which is equivalent to $x$ and $hat y-x$ being orthogonal. This should remind you of Thales's theorem, as the set of all such $x$ is the sphere with diameter the line segment from $0$ to $hat y$. That is, it is the set of all $x$ satisfying
                      $$
                      left| x - frac{hat y}{2}right| = left| frac{hat y}{2}right|
                      $$

                      or in the original variables
                      $$
                      left| x - frac{y}{2langle y,yrangle}right| = left| frac{y}{2langle y,yrangle}right| = frac{1}{2|y|}.
                      $$



                      In summary for $y=0$ every $x$ is possible, for each $yneq 0$ you have all $x$ on the sphere with diameter the line segment from $0$ to $y/langle y,yrangle$.






                      share|cite|improve this answer











                      $endgroup$













                      • $begingroup$
                        You really mean sphere not circle ($n$ is not necessarily $1$), and this is another way of phrasing it.
                        $endgroup$
                        – user10354138
                        Dec 20 '18 at 14:19










                      • $begingroup$
                        You are right, thanks. I was thinking about $n=2$. Good thing Thales's theorem holds for arbitrary spheres ;-)
                        $endgroup$
                        – Christoph
                        Dec 20 '18 at 15:46
















                      0












                      $begingroup$

                      The pair $(x,y)$ satisfies $langle x,yrangle = langle x,xranglelangle y,yrangle$ when $y=0$. When $yneq 0$ we may set $hat y = y/langle y,yrangle$ to obtain the equation $langle x,hat y rangle = langle x,xrangle$ which is equivalent to $x$ and $hat y-x$ being orthogonal. This should remind you of Thales's theorem, as the set of all such $x$ is the sphere with diameter the line segment from $0$ to $hat y$. That is, it is the set of all $x$ satisfying
                      $$
                      left| x - frac{hat y}{2}right| = left| frac{hat y}{2}right|
                      $$

                      or in the original variables
                      $$
                      left| x - frac{y}{2langle y,yrangle}right| = left| frac{y}{2langle y,yrangle}right| = frac{1}{2|y|}.
                      $$



                      In summary for $y=0$ every $x$ is possible, for each $yneq 0$ you have all $x$ on the sphere with diameter the line segment from $0$ to $y/langle y,yrangle$.






                      share|cite|improve this answer











                      $endgroup$













                      • $begingroup$
                        You really mean sphere not circle ($n$ is not necessarily $1$), and this is another way of phrasing it.
                        $endgroup$
                        – user10354138
                        Dec 20 '18 at 14:19










                      • $begingroup$
                        You are right, thanks. I was thinking about $n=2$. Good thing Thales's theorem holds for arbitrary spheres ;-)
                        $endgroup$
                        – Christoph
                        Dec 20 '18 at 15:46














                      0












                      0








                      0





                      $begingroup$

                      The pair $(x,y)$ satisfies $langle x,yrangle = langle x,xranglelangle y,yrangle$ when $y=0$. When $yneq 0$ we may set $hat y = y/langle y,yrangle$ to obtain the equation $langle x,hat y rangle = langle x,xrangle$ which is equivalent to $x$ and $hat y-x$ being orthogonal. This should remind you of Thales's theorem, as the set of all such $x$ is the sphere with diameter the line segment from $0$ to $hat y$. That is, it is the set of all $x$ satisfying
                      $$
                      left| x - frac{hat y}{2}right| = left| frac{hat y}{2}right|
                      $$

                      or in the original variables
                      $$
                      left| x - frac{y}{2langle y,yrangle}right| = left| frac{y}{2langle y,yrangle}right| = frac{1}{2|y|}.
                      $$



                      In summary for $y=0$ every $x$ is possible, for each $yneq 0$ you have all $x$ on the sphere with diameter the line segment from $0$ to $y/langle y,yrangle$.






                      share|cite|improve this answer











                      $endgroup$



                      The pair $(x,y)$ satisfies $langle x,yrangle = langle x,xranglelangle y,yrangle$ when $y=0$. When $yneq 0$ we may set $hat y = y/langle y,yrangle$ to obtain the equation $langle x,hat y rangle = langle x,xrangle$ which is equivalent to $x$ and $hat y-x$ being orthogonal. This should remind you of Thales's theorem, as the set of all such $x$ is the sphere with diameter the line segment from $0$ to $hat y$. That is, it is the set of all $x$ satisfying
                      $$
                      left| x - frac{hat y}{2}right| = left| frac{hat y}{2}right|
                      $$

                      or in the original variables
                      $$
                      left| x - frac{y}{2langle y,yrangle}right| = left| frac{y}{2langle y,yrangle}right| = frac{1}{2|y|}.
                      $$



                      In summary for $y=0$ every $x$ is possible, for each $yneq 0$ you have all $x$ on the sphere with diameter the line segment from $0$ to $y/langle y,yrangle$.







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited Dec 20 '18 at 15:44

























                      answered Dec 20 '18 at 12:44









                      ChristophChristoph

                      12.3k1642




                      12.3k1642












                      • $begingroup$
                        You really mean sphere not circle ($n$ is not necessarily $1$), and this is another way of phrasing it.
                        $endgroup$
                        – user10354138
                        Dec 20 '18 at 14:19










                      • $begingroup$
                        You are right, thanks. I was thinking about $n=2$. Good thing Thales's theorem holds for arbitrary spheres ;-)
                        $endgroup$
                        – Christoph
                        Dec 20 '18 at 15:46


















                      • $begingroup$
                        You really mean sphere not circle ($n$ is not necessarily $1$), and this is another way of phrasing it.
                        $endgroup$
                        – user10354138
                        Dec 20 '18 at 14:19










                      • $begingroup$
                        You are right, thanks. I was thinking about $n=2$. Good thing Thales's theorem holds for arbitrary spheres ;-)
                        $endgroup$
                        – Christoph
                        Dec 20 '18 at 15:46
















                      $begingroup$
                      You really mean sphere not circle ($n$ is not necessarily $1$), and this is another way of phrasing it.
                      $endgroup$
                      – user10354138
                      Dec 20 '18 at 14:19




                      $begingroup$
                      You really mean sphere not circle ($n$ is not necessarily $1$), and this is another way of phrasing it.
                      $endgroup$
                      – user10354138
                      Dec 20 '18 at 14:19












                      $begingroup$
                      You are right, thanks. I was thinking about $n=2$. Good thing Thales's theorem holds for arbitrary spheres ;-)
                      $endgroup$
                      – Christoph
                      Dec 20 '18 at 15:46




                      $begingroup$
                      You are right, thanks. I was thinking about $n=2$. Good thing Thales's theorem holds for arbitrary spheres ;-)
                      $endgroup$
                      – Christoph
                      Dec 20 '18 at 15:46


















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