Prove that linear combination of two 3D independent vector will form a plane rather than a Sold

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How could we prove that linear combination of two 3D independent vector will always form a plane rather than a Solid?



Regards,
Tarun







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    How could we prove that linear combination of two 3D independent vector will always form a plane rather than a Solid?



    Regards,
    Tarun







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      up vote
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      favorite









      up vote
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      down vote

      favorite











      How could we prove that linear combination of two 3D independent vector will always form a plane rather than a Solid?



      Regards,
      Tarun







      share|cite|improve this question












      How could we prove that linear combination of two 3D independent vector will always form a plane rather than a Solid?



      Regards,
      Tarun









      share|cite|improve this question











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      asked Aug 24 at 9:01









      fluty

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          I can see two ways to answer your question.



          Definition of a plane



          A (vector) plane is by definition the set of linear combinations of two independent vectors.



          A plane doesn't contain an open ball



          It's again a question of definition. How do you define a solid? If you mean by a solid a subset of $mathbb R^3$ which contains at least one open ball ("it should be able to contain some volume"), then you can prove that an open ball can't be contained in a plane.






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          • I was having difficulty to imagine how linear combination of two vector will always form a plane in 3 dimension. Yes by definition of plane it is true that it will always form a plane
            – fluty
            Aug 24 at 9:21










          Your Answer




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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          2
          down vote













          I can see two ways to answer your question.



          Definition of a plane



          A (vector) plane is by definition the set of linear combinations of two independent vectors.



          A plane doesn't contain an open ball



          It's again a question of definition. How do you define a solid? If you mean by a solid a subset of $mathbb R^3$ which contains at least one open ball ("it should be able to contain some volume"), then you can prove that an open ball can't be contained in a plane.






          share|cite|improve this answer




















          • I was having difficulty to imagine how linear combination of two vector will always form a plane in 3 dimension. Yes by definition of plane it is true that it will always form a plane
            – fluty
            Aug 24 at 9:21














          up vote
          2
          down vote













          I can see two ways to answer your question.



          Definition of a plane



          A (vector) plane is by definition the set of linear combinations of two independent vectors.



          A plane doesn't contain an open ball



          It's again a question of definition. How do you define a solid? If you mean by a solid a subset of $mathbb R^3$ which contains at least one open ball ("it should be able to contain some volume"), then you can prove that an open ball can't be contained in a plane.






          share|cite|improve this answer




















          • I was having difficulty to imagine how linear combination of two vector will always form a plane in 3 dimension. Yes by definition of plane it is true that it will always form a plane
            – fluty
            Aug 24 at 9:21












          up vote
          2
          down vote










          up vote
          2
          down vote









          I can see two ways to answer your question.



          Definition of a plane



          A (vector) plane is by definition the set of linear combinations of two independent vectors.



          A plane doesn't contain an open ball



          It's again a question of definition. How do you define a solid? If you mean by a solid a subset of $mathbb R^3$ which contains at least one open ball ("it should be able to contain some volume"), then you can prove that an open ball can't be contained in a plane.






          share|cite|improve this answer












          I can see two ways to answer your question.



          Definition of a plane



          A (vector) plane is by definition the set of linear combinations of two independent vectors.



          A plane doesn't contain an open ball



          It's again a question of definition. How do you define a solid? If you mean by a solid a subset of $mathbb R^3$ which contains at least one open ball ("it should be able to contain some volume"), then you can prove that an open ball can't be contained in a plane.







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          share|cite|improve this answer



          share|cite|improve this answer










          answered Aug 24 at 9:10









          mathcounterexamples.net

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          • I was having difficulty to imagine how linear combination of two vector will always form a plane in 3 dimension. Yes by definition of plane it is true that it will always form a plane
            – fluty
            Aug 24 at 9:21
















          • I was having difficulty to imagine how linear combination of two vector will always form a plane in 3 dimension. Yes by definition of plane it is true that it will always form a plane
            – fluty
            Aug 24 at 9:21















          I was having difficulty to imagine how linear combination of two vector will always form a plane in 3 dimension. Yes by definition of plane it is true that it will always form a plane
          – fluty
          Aug 24 at 9:21




          I was having difficulty to imagine how linear combination of two vector will always form a plane in 3 dimension. Yes by definition of plane it is true that it will always form a plane
          – fluty
          Aug 24 at 9:21

















           

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