What is an orthonormal basis of a plane?

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I have a plane $ax+by+cz+d=0$, what is an orthonormal basis for it?



Is it a set of three 3D vectors $e_1, e_2, e_3$ with the following properties?



  1. each $e_i$ has unit length;


  2. $e_1perp e_2$, $e_2perp e_3$, $e_3perp e_1$;


  3. $e_1$ and $e_2$ lie on the plane and so $e_3$ is parallel to $(a,b,c)$.





linear-algebra geometry







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    up vote
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    down vote

    favorite












    I have a plane $ax+by+cz+d=0$, what is an orthonormal basis for it?



    Is it a set of three 3D vectors $e_1, e_2, e_3$ with the following properties?



    1. each $e_i$ has unit length;


    2. $e_1perp e_2$, $e_2perp e_3$, $e_3perp e_1$;


    3. $e_1$ and $e_2$ lie on the plane and so $e_3$ is parallel to $(a,b,c)$.





    linear-algebra geometry







    share|cite|improve this question























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I have a plane $ax+by+cz+d=0$, what is an orthonormal basis for it?



      Is it a set of three 3D vectors $e_1, e_2, e_3$ with the following properties?



      1. each $e_i$ has unit length;


      2. $e_1perp e_2$, $e_2perp e_3$, $e_3perp e_1$;


      3. $e_1$ and $e_2$ lie on the plane and so $e_3$ is parallel to $(a,b,c)$.





      linear-algebra geometry







      share|cite|improve this question













      I have a plane $ax+by+cz+d=0$, what is an orthonormal basis for it?



      Is it a set of three 3D vectors $e_1, e_2, e_3$ with the following properties?



      1. each $e_i$ has unit length;


      2. $e_1perp e_2$, $e_2perp e_3$, $e_3perp e_1$;


      3. $e_1$ and $e_2$ lie on the plane and so $e_3$ is parallel to $(a,b,c)$.





      linear-algebra geometry




      linear-algebra geometry






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      asked 1 hour ago









      Alessandro Jacopson

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          A plane is a two dimensional object, so the basis will have only two elements (two 3D vectors). In your notation $e_3$ is not part of the basis. Everything else is fine.






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            You can state all of the vectors in a plane by just two none-zeeo vectors in different directions. The set of these two vectors is called a basis for plane vector space. As you see, there are infinite bases for a plane!
            Orthogonality and orthonormality can be difined only after defining norms. Otherwise each basis see itself orthonormal, because in it's point of view the base vectors in it are normal (unit lenght) and orthogonal to each other!
            Maybe it's some strange but for understanding that you can think about seeing object from a magnifying-glass!






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              2 Answers
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              2 Answers
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              up vote
              4
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              A plane is a two dimensional object, so the basis will have only two elements (two 3D vectors). In your notation $e_3$ is not part of the basis. Everything else is fine.






              share|cite|improve this answer
























                up vote
                4
                down vote













                A plane is a two dimensional object, so the basis will have only two elements (two 3D vectors). In your notation $e_3$ is not part of the basis. Everything else is fine.






                share|cite|improve this answer






















                  up vote
                  4
                  down vote










                  up vote
                  4
                  down vote









                  A plane is a two dimensional object, so the basis will have only two elements (two 3D vectors). In your notation $e_3$ is not part of the basis. Everything else is fine.






                  share|cite|improve this answer












                  A plane is a two dimensional object, so the basis will have only two elements (two 3D vectors). In your notation $e_3$ is not part of the basis. Everything else is fine.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 1 hour ago









                  Andrei

                  8,4432923




                  8,4432923




















                      up vote
                      2
                      down vote













                      You can state all of the vectors in a plane by just two none-zeeo vectors in different directions. The set of these two vectors is called a basis for plane vector space. As you see, there are infinite bases for a plane!
                      Orthogonality and orthonormality can be difined only after defining norms. Otherwise each basis see itself orthonormal, because in it's point of view the base vectors in it are normal (unit lenght) and orthogonal to each other!
                      Maybe it's some strange but for understanding that you can think about seeing object from a magnifying-glass!






                      share|cite|improve this answer
























                        up vote
                        2
                        down vote













                        You can state all of the vectors in a plane by just two none-zeeo vectors in different directions. The set of these two vectors is called a basis for plane vector space. As you see, there are infinite bases for a plane!
                        Orthogonality and orthonormality can be difined only after defining norms. Otherwise each basis see itself orthonormal, because in it's point of view the base vectors in it are normal (unit lenght) and orthogonal to each other!
                        Maybe it's some strange but for understanding that you can think about seeing object from a magnifying-glass!






                        share|cite|improve this answer






















                          up vote
                          2
                          down vote










                          up vote
                          2
                          down vote









                          You can state all of the vectors in a plane by just two none-zeeo vectors in different directions. The set of these two vectors is called a basis for plane vector space. As you see, there are infinite bases for a plane!
                          Orthogonality and orthonormality can be difined only after defining norms. Otherwise each basis see itself orthonormal, because in it's point of view the base vectors in it are normal (unit lenght) and orthogonal to each other!
                          Maybe it's some strange but for understanding that you can think about seeing object from a magnifying-glass!






                          share|cite|improve this answer












                          You can state all of the vectors in a plane by just two none-zeeo vectors in different directions. The set of these two vectors is called a basis for plane vector space. As you see, there are infinite bases for a plane!
                          Orthogonality and orthonormality can be difined only after defining norms. Otherwise each basis see itself orthonormal, because in it's point of view the base vectors in it are normal (unit lenght) and orthogonal to each other!
                          Maybe it's some strange but for understanding that you can think about seeing object from a magnifying-glass!







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 36 mins ago









                          Hossein Sharif

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