Sum of squared inner product of vector with spokes around unit circle is constant












4














Let $v$ be any vector in the plane, and ${w_i}$ be $n>2$ vectors evenly spaced around the unit circle. Then it seems true that
$$sum_{i=1}^n (vcdot w_i)^2 = k |v|^2$$
where $k$ is a constant independent of $v$. My intuition is that the left-hand side, as a function of the argument $theta$ of $v$, has "too much symmetry" for its low frequency.



Is there some slick proof of this identity, using e.g. the algebraic structure of the $n$ roots of unity?










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    4














    Let $v$ be any vector in the plane, and ${w_i}$ be $n>2$ vectors evenly spaced around the unit circle. Then it seems true that
    $$sum_{i=1}^n (vcdot w_i)^2 = k |v|^2$$
    where $k$ is a constant independent of $v$. My intuition is that the left-hand side, as a function of the argument $theta$ of $v$, has "too much symmetry" for its low frequency.



    Is there some slick proof of this identity, using e.g. the algebraic structure of the $n$ roots of unity?










    share|cite|improve this question

























      4












      4








      4







      Let $v$ be any vector in the plane, and ${w_i}$ be $n>2$ vectors evenly spaced around the unit circle. Then it seems true that
      $$sum_{i=1}^n (vcdot w_i)^2 = k |v|^2$$
      where $k$ is a constant independent of $v$. My intuition is that the left-hand side, as a function of the argument $theta$ of $v$, has "too much symmetry" for its low frequency.



      Is there some slick proof of this identity, using e.g. the algebraic structure of the $n$ roots of unity?










      share|cite|improve this question













      Let $v$ be any vector in the plane, and ${w_i}$ be $n>2$ vectors evenly spaced around the unit circle. Then it seems true that
      $$sum_{i=1}^n (vcdot w_i)^2 = k |v|^2$$
      where $k$ is a constant independent of $v$. My intuition is that the left-hand side, as a function of the argument $theta$ of $v$, has "too much symmetry" for its low frequency.



      Is there some slick proof of this identity, using e.g. the algebraic structure of the $n$ roots of unity?







      linear-algebra plane-geometry






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      asked 1 hour ago









      user7530

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      34.5k759113






















          2 Answers
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          3














          This is similar to Noble's answer but I wanted to highlight the geometric aspect of the problem a little more. First we note that
          $$v cdot w_i = || v || cos(theta) $$
          where $theta$ is the angle between $v$ and $w_i$. Since $||v||^2$ is on both sides of the equation, we can assume $||v|| = 1$.



          Now consider a unit vector $v$ and a regular polygon with $n$ sides centered at the origin. Connect the vertices of the polygon to the center. $v$ lies between two of the radii. Now going counter clockwise, $v$ makes the following angles with the $w_i:$
          $$x, x + frac{2 pi}n, x + 2 cdot frac{2 pi}n, cdots, x + (n-1) cdot frac{2 pi}n $$
          for some $x$ as shown in the image at the bottom of the post.



          Therefore, the sum that we want is
          $$ sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2.$$



          Using Euler's formula $cos(t) = frac{e^{it} + e^{-it}}2$, we have



          begin{align}
          sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2 &= frac{1}4 sum_{k=0}^{n-1} left(e^{2ileft(x + k, frac{2pi}n right)} + e^{-2ileft(x + k, frac{2pi}n right)} + 2right) \
          &= frac{n}2 + (e^{2ix} + e^{-2ix}) sum_{k=0}^{n-1} omega^k
          end{align}

          where $omega$ is a non trivial $n$th root of unity. Then we can easily calculate that
          $$ sum_{k=0}^{n-1} omega^k = frac{omega^n-1}{omega - 1} = 0$$
          which proves the claim for $k = frac{n}2.$



          Angles between v and w






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            2














            Since the $w_i$ are evenly spaced, we can say:



            $$w_i=cos(phi+itheta)hat i+sin(phi+itheta)hat j$$



            Also, since they are evenly spaced around the full circle, the distances between each $w_i$, which is $theta$ radians, times the number of vectors, which is $n$, must be a full circle: That is, $ntheta=2pi$



            Now, $vcdot w_i=v_xcos(phi+itheta)+v_ysin(phi+itheta)$, so:



            $$(vcdot w_i)^2=v_x^2cos^2(phi+itheta)+v_y^2sin^2(phi+itheta)+2v_xv_ycos(phi+itheta)sin(phi+itheta) \ =v_x^2left(frac{cos(2phi+2itheta)+1}{2}right)+v_y^2left(frac{1-cos(2phi+2itheta)}{2}right)+v_xv_ysin(2phi+2itheta) \ =frac{v_x^2+v_y^2}{2}+cos(2phi+2itheta)left(frac{v_x^2-v_y^2}{2}right)+v_xv_ysin(2phi+2itheta)$$



            Now, let's put this into a summation:



            $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)left[sum_{i=1}^ncos(2phi+2itheta)right]+v_xv_yleft[sum_{i=1}^nsin(2phi+2itheta)right]$$



            Now, I will use $j=sqrt{-1}$. The first summation and second summation are related: They are the real and imaginary part, respectively, of $sum_{i=1}^n e^{2jphi+2ijtheta}=e^{2jphi}sum_{i=1}^n e^{2ijtheta}$. If we let $omega=e^{2jtheta}$, the sum becomes $e^{2jphi}sum_{i=1}^n omega^i$, which is a geometric series:



            $$sum_{i=1}^n omega^i=omegafrac{omega^n-1}{omega-1}=omegafrac{e^{2jntheta}-1}{omega-1}=omegafrac{e^{4pi j}-1}{omega-1}=omegafrac{1-1}{omega-1}=0$$



            (Note that I used $ntheta=2pi$ and $e^{4pi j}=cos(4pi)+jsin(4pi)=1$ in the above derivation.)



            (Also, notice that this proof fails when $omega=e^{2jtheta}=1$ because of a division-by-zero error from above. More specifically, this occurs when $2theta=frac{4pi}{n}$ is a multiple of $2pi$, which is the case when there are $n=1$ or $n=2$ points.)



            Thus, this summation is $0$, so both its real and imaginary parts are 0. This gives us:



            $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)[0]+v_xv_y[0]rightarrow sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}|v|^2$$






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              2 Answers
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              3














              This is similar to Noble's answer but I wanted to highlight the geometric aspect of the problem a little more. First we note that
              $$v cdot w_i = || v || cos(theta) $$
              where $theta$ is the angle between $v$ and $w_i$. Since $||v||^2$ is on both sides of the equation, we can assume $||v|| = 1$.



              Now consider a unit vector $v$ and a regular polygon with $n$ sides centered at the origin. Connect the vertices of the polygon to the center. $v$ lies between two of the radii. Now going counter clockwise, $v$ makes the following angles with the $w_i:$
              $$x, x + frac{2 pi}n, x + 2 cdot frac{2 pi}n, cdots, x + (n-1) cdot frac{2 pi}n $$
              for some $x$ as shown in the image at the bottom of the post.



              Therefore, the sum that we want is
              $$ sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2.$$



              Using Euler's formula $cos(t) = frac{e^{it} + e^{-it}}2$, we have



              begin{align}
              sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2 &= frac{1}4 sum_{k=0}^{n-1} left(e^{2ileft(x + k, frac{2pi}n right)} + e^{-2ileft(x + k, frac{2pi}n right)} + 2right) \
              &= frac{n}2 + (e^{2ix} + e^{-2ix}) sum_{k=0}^{n-1} omega^k
              end{align}

              where $omega$ is a non trivial $n$th root of unity. Then we can easily calculate that
              $$ sum_{k=0}^{n-1} omega^k = frac{omega^n-1}{omega - 1} = 0$$
              which proves the claim for $k = frac{n}2.$



              Angles between v and w






              share|cite|improve this answer


























                3














                This is similar to Noble's answer but I wanted to highlight the geometric aspect of the problem a little more. First we note that
                $$v cdot w_i = || v || cos(theta) $$
                where $theta$ is the angle between $v$ and $w_i$. Since $||v||^2$ is on both sides of the equation, we can assume $||v|| = 1$.



                Now consider a unit vector $v$ and a regular polygon with $n$ sides centered at the origin. Connect the vertices of the polygon to the center. $v$ lies between two of the radii. Now going counter clockwise, $v$ makes the following angles with the $w_i:$
                $$x, x + frac{2 pi}n, x + 2 cdot frac{2 pi}n, cdots, x + (n-1) cdot frac{2 pi}n $$
                for some $x$ as shown in the image at the bottom of the post.



                Therefore, the sum that we want is
                $$ sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2.$$



                Using Euler's formula $cos(t) = frac{e^{it} + e^{-it}}2$, we have



                begin{align}
                sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2 &= frac{1}4 sum_{k=0}^{n-1} left(e^{2ileft(x + k, frac{2pi}n right)} + e^{-2ileft(x + k, frac{2pi}n right)} + 2right) \
                &= frac{n}2 + (e^{2ix} + e^{-2ix}) sum_{k=0}^{n-1} omega^k
                end{align}

                where $omega$ is a non trivial $n$th root of unity. Then we can easily calculate that
                $$ sum_{k=0}^{n-1} omega^k = frac{omega^n-1}{omega - 1} = 0$$
                which proves the claim for $k = frac{n}2.$



                Angles between v and w






                share|cite|improve this answer
























                  3












                  3








                  3






                  This is similar to Noble's answer but I wanted to highlight the geometric aspect of the problem a little more. First we note that
                  $$v cdot w_i = || v || cos(theta) $$
                  where $theta$ is the angle between $v$ and $w_i$. Since $||v||^2$ is on both sides of the equation, we can assume $||v|| = 1$.



                  Now consider a unit vector $v$ and a regular polygon with $n$ sides centered at the origin. Connect the vertices of the polygon to the center. $v$ lies between two of the radii. Now going counter clockwise, $v$ makes the following angles with the $w_i:$
                  $$x, x + frac{2 pi}n, x + 2 cdot frac{2 pi}n, cdots, x + (n-1) cdot frac{2 pi}n $$
                  for some $x$ as shown in the image at the bottom of the post.



                  Therefore, the sum that we want is
                  $$ sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2.$$



                  Using Euler's formula $cos(t) = frac{e^{it} + e^{-it}}2$, we have



                  begin{align}
                  sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2 &= frac{1}4 sum_{k=0}^{n-1} left(e^{2ileft(x + k, frac{2pi}n right)} + e^{-2ileft(x + k, frac{2pi}n right)} + 2right) \
                  &= frac{n}2 + (e^{2ix} + e^{-2ix}) sum_{k=0}^{n-1} omega^k
                  end{align}

                  where $omega$ is a non trivial $n$th root of unity. Then we can easily calculate that
                  $$ sum_{k=0}^{n-1} omega^k = frac{omega^n-1}{omega - 1} = 0$$
                  which proves the claim for $k = frac{n}2.$



                  Angles between v and w






                  share|cite|improve this answer












                  This is similar to Noble's answer but I wanted to highlight the geometric aspect of the problem a little more. First we note that
                  $$v cdot w_i = || v || cos(theta) $$
                  where $theta$ is the angle between $v$ and $w_i$. Since $||v||^2$ is on both sides of the equation, we can assume $||v|| = 1$.



                  Now consider a unit vector $v$ and a regular polygon with $n$ sides centered at the origin. Connect the vertices of the polygon to the center. $v$ lies between two of the radii. Now going counter clockwise, $v$ makes the following angles with the $w_i:$
                  $$x, x + frac{2 pi}n, x + 2 cdot frac{2 pi}n, cdots, x + (n-1) cdot frac{2 pi}n $$
                  for some $x$ as shown in the image at the bottom of the post.



                  Therefore, the sum that we want is
                  $$ sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2.$$



                  Using Euler's formula $cos(t) = frac{e^{it} + e^{-it}}2$, we have



                  begin{align}
                  sum_{k=0}^{n-1} cos left( x + k , frac{2 pi}n right)^2 &= frac{1}4 sum_{k=0}^{n-1} left(e^{2ileft(x + k, frac{2pi}n right)} + e^{-2ileft(x + k, frac{2pi}n right)} + 2right) \
                  &= frac{n}2 + (e^{2ix} + e^{-2ix}) sum_{k=0}^{n-1} omega^k
                  end{align}

                  where $omega$ is a non trivial $n$th root of unity. Then we can easily calculate that
                  $$ sum_{k=0}^{n-1} omega^k = frac{omega^n-1}{omega - 1} = 0$$
                  which proves the claim for $k = frac{n}2.$



                  Angles between v and w







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                  answered 52 mins ago









                  Sandeep Silwal

                  5,68011236




                  5,68011236























                      2














                      Since the $w_i$ are evenly spaced, we can say:



                      $$w_i=cos(phi+itheta)hat i+sin(phi+itheta)hat j$$



                      Also, since they are evenly spaced around the full circle, the distances between each $w_i$, which is $theta$ radians, times the number of vectors, which is $n$, must be a full circle: That is, $ntheta=2pi$



                      Now, $vcdot w_i=v_xcos(phi+itheta)+v_ysin(phi+itheta)$, so:



                      $$(vcdot w_i)^2=v_x^2cos^2(phi+itheta)+v_y^2sin^2(phi+itheta)+2v_xv_ycos(phi+itheta)sin(phi+itheta) \ =v_x^2left(frac{cos(2phi+2itheta)+1}{2}right)+v_y^2left(frac{1-cos(2phi+2itheta)}{2}right)+v_xv_ysin(2phi+2itheta) \ =frac{v_x^2+v_y^2}{2}+cos(2phi+2itheta)left(frac{v_x^2-v_y^2}{2}right)+v_xv_ysin(2phi+2itheta)$$



                      Now, let's put this into a summation:



                      $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)left[sum_{i=1}^ncos(2phi+2itheta)right]+v_xv_yleft[sum_{i=1}^nsin(2phi+2itheta)right]$$



                      Now, I will use $j=sqrt{-1}$. The first summation and second summation are related: They are the real and imaginary part, respectively, of $sum_{i=1}^n e^{2jphi+2ijtheta}=e^{2jphi}sum_{i=1}^n e^{2ijtheta}$. If we let $omega=e^{2jtheta}$, the sum becomes $e^{2jphi}sum_{i=1}^n omega^i$, which is a geometric series:



                      $$sum_{i=1}^n omega^i=omegafrac{omega^n-1}{omega-1}=omegafrac{e^{2jntheta}-1}{omega-1}=omegafrac{e^{4pi j}-1}{omega-1}=omegafrac{1-1}{omega-1}=0$$



                      (Note that I used $ntheta=2pi$ and $e^{4pi j}=cos(4pi)+jsin(4pi)=1$ in the above derivation.)



                      (Also, notice that this proof fails when $omega=e^{2jtheta}=1$ because of a division-by-zero error from above. More specifically, this occurs when $2theta=frac{4pi}{n}$ is a multiple of $2pi$, which is the case when there are $n=1$ or $n=2$ points.)



                      Thus, this summation is $0$, so both its real and imaginary parts are 0. This gives us:



                      $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)[0]+v_xv_y[0]rightarrow sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}|v|^2$$






                      share|cite|improve this answer




























                        2














                        Since the $w_i$ are evenly spaced, we can say:



                        $$w_i=cos(phi+itheta)hat i+sin(phi+itheta)hat j$$



                        Also, since they are evenly spaced around the full circle, the distances between each $w_i$, which is $theta$ radians, times the number of vectors, which is $n$, must be a full circle: That is, $ntheta=2pi$



                        Now, $vcdot w_i=v_xcos(phi+itheta)+v_ysin(phi+itheta)$, so:



                        $$(vcdot w_i)^2=v_x^2cos^2(phi+itheta)+v_y^2sin^2(phi+itheta)+2v_xv_ycos(phi+itheta)sin(phi+itheta) \ =v_x^2left(frac{cos(2phi+2itheta)+1}{2}right)+v_y^2left(frac{1-cos(2phi+2itheta)}{2}right)+v_xv_ysin(2phi+2itheta) \ =frac{v_x^2+v_y^2}{2}+cos(2phi+2itheta)left(frac{v_x^2-v_y^2}{2}right)+v_xv_ysin(2phi+2itheta)$$



                        Now, let's put this into a summation:



                        $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)left[sum_{i=1}^ncos(2phi+2itheta)right]+v_xv_yleft[sum_{i=1}^nsin(2phi+2itheta)right]$$



                        Now, I will use $j=sqrt{-1}$. The first summation and second summation are related: They are the real and imaginary part, respectively, of $sum_{i=1}^n e^{2jphi+2ijtheta}=e^{2jphi}sum_{i=1}^n e^{2ijtheta}$. If we let $omega=e^{2jtheta}$, the sum becomes $e^{2jphi}sum_{i=1}^n omega^i$, which is a geometric series:



                        $$sum_{i=1}^n omega^i=omegafrac{omega^n-1}{omega-1}=omegafrac{e^{2jntheta}-1}{omega-1}=omegafrac{e^{4pi j}-1}{omega-1}=omegafrac{1-1}{omega-1}=0$$



                        (Note that I used $ntheta=2pi$ and $e^{4pi j}=cos(4pi)+jsin(4pi)=1$ in the above derivation.)



                        (Also, notice that this proof fails when $omega=e^{2jtheta}=1$ because of a division-by-zero error from above. More specifically, this occurs when $2theta=frac{4pi}{n}$ is a multiple of $2pi$, which is the case when there are $n=1$ or $n=2$ points.)



                        Thus, this summation is $0$, so both its real and imaginary parts are 0. This gives us:



                        $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)[0]+v_xv_y[0]rightarrow sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}|v|^2$$






                        share|cite|improve this answer


























                          2












                          2








                          2






                          Since the $w_i$ are evenly spaced, we can say:



                          $$w_i=cos(phi+itheta)hat i+sin(phi+itheta)hat j$$



                          Also, since they are evenly spaced around the full circle, the distances between each $w_i$, which is $theta$ radians, times the number of vectors, which is $n$, must be a full circle: That is, $ntheta=2pi$



                          Now, $vcdot w_i=v_xcos(phi+itheta)+v_ysin(phi+itheta)$, so:



                          $$(vcdot w_i)^2=v_x^2cos^2(phi+itheta)+v_y^2sin^2(phi+itheta)+2v_xv_ycos(phi+itheta)sin(phi+itheta) \ =v_x^2left(frac{cos(2phi+2itheta)+1}{2}right)+v_y^2left(frac{1-cos(2phi+2itheta)}{2}right)+v_xv_ysin(2phi+2itheta) \ =frac{v_x^2+v_y^2}{2}+cos(2phi+2itheta)left(frac{v_x^2-v_y^2}{2}right)+v_xv_ysin(2phi+2itheta)$$



                          Now, let's put this into a summation:



                          $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)left[sum_{i=1}^ncos(2phi+2itheta)right]+v_xv_yleft[sum_{i=1}^nsin(2phi+2itheta)right]$$



                          Now, I will use $j=sqrt{-1}$. The first summation and second summation are related: They are the real and imaginary part, respectively, of $sum_{i=1}^n e^{2jphi+2ijtheta}=e^{2jphi}sum_{i=1}^n e^{2ijtheta}$. If we let $omega=e^{2jtheta}$, the sum becomes $e^{2jphi}sum_{i=1}^n omega^i$, which is a geometric series:



                          $$sum_{i=1}^n omega^i=omegafrac{omega^n-1}{omega-1}=omegafrac{e^{2jntheta}-1}{omega-1}=omegafrac{e^{4pi j}-1}{omega-1}=omegafrac{1-1}{omega-1}=0$$



                          (Note that I used $ntheta=2pi$ and $e^{4pi j}=cos(4pi)+jsin(4pi)=1$ in the above derivation.)



                          (Also, notice that this proof fails when $omega=e^{2jtheta}=1$ because of a division-by-zero error from above. More specifically, this occurs when $2theta=frac{4pi}{n}$ is a multiple of $2pi$, which is the case when there are $n=1$ or $n=2$ points.)



                          Thus, this summation is $0$, so both its real and imaginary parts are 0. This gives us:



                          $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)[0]+v_xv_y[0]rightarrow sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}|v|^2$$






                          share|cite|improve this answer














                          Since the $w_i$ are evenly spaced, we can say:



                          $$w_i=cos(phi+itheta)hat i+sin(phi+itheta)hat j$$



                          Also, since they are evenly spaced around the full circle, the distances between each $w_i$, which is $theta$ radians, times the number of vectors, which is $n$, must be a full circle: That is, $ntheta=2pi$



                          Now, $vcdot w_i=v_xcos(phi+itheta)+v_ysin(phi+itheta)$, so:



                          $$(vcdot w_i)^2=v_x^2cos^2(phi+itheta)+v_y^2sin^2(phi+itheta)+2v_xv_ycos(phi+itheta)sin(phi+itheta) \ =v_x^2left(frac{cos(2phi+2itheta)+1}{2}right)+v_y^2left(frac{1-cos(2phi+2itheta)}{2}right)+v_xv_ysin(2phi+2itheta) \ =frac{v_x^2+v_y^2}{2}+cos(2phi+2itheta)left(frac{v_x^2-v_y^2}{2}right)+v_xv_ysin(2phi+2itheta)$$



                          Now, let's put this into a summation:



                          $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)left[sum_{i=1}^ncos(2phi+2itheta)right]+v_xv_yleft[sum_{i=1}^nsin(2phi+2itheta)right]$$



                          Now, I will use $j=sqrt{-1}$. The first summation and second summation are related: They are the real and imaginary part, respectively, of $sum_{i=1}^n e^{2jphi+2ijtheta}=e^{2jphi}sum_{i=1}^n e^{2ijtheta}$. If we let $omega=e^{2jtheta}$, the sum becomes $e^{2jphi}sum_{i=1}^n omega^i$, which is a geometric series:



                          $$sum_{i=1}^n omega^i=omegafrac{omega^n-1}{omega-1}=omegafrac{e^{2jntheta}-1}{omega-1}=omegafrac{e^{4pi j}-1}{omega-1}=omegafrac{1-1}{omega-1}=0$$



                          (Note that I used $ntheta=2pi$ and $e^{4pi j}=cos(4pi)+jsin(4pi)=1$ in the above derivation.)



                          (Also, notice that this proof fails when $omega=e^{2jtheta}=1$ because of a division-by-zero error from above. More specifically, this occurs when $2theta=frac{4pi}{n}$ is a multiple of $2pi$, which is the case when there are $n=1$ or $n=2$ points.)



                          Thus, this summation is $0$, so both its real and imaginary parts are 0. This gives us:



                          $$sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}(v_x^2+v_y^2)+left(frac{v_x^2-v_y^2}{2}right)[0]+v_xv_y[0]rightarrow sum_{i=1}^n (vcdot w_i)^2=frac{n}{2}|v|^2$$







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                          edited 1 hour ago

























                          answered 1 hour ago









                          Noble Mushtak

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