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
linear-algebra vectors 3d
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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
linear-algebra vectors 3d
add a comment |Â
up vote
1
down vote
favorite
up vote
1
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
linear-algebra vectors 3d
How could we prove that linear combination of two 3D independent vector will always form a plane rather than a Solid?
Regards,
Tarun
linear-algebra vectors 3d
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.
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
â fluty
Aug 24 at 9:21
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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.
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
â fluty
Aug 24 at 9:21
add a comment |Â
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.
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
â fluty
Aug 24 at 9:21
add a comment |Â
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.
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.
answered Aug 24 at 9:10
mathcounterexamples.net
25.5k21755
25.5k21755
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
â fluty
Aug 24 at 9:21
add a comment |Â
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
â 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
â 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
â fluty
Aug 24 at 9:21
add a comment |Â
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