How could we prove that linear combination of two 3D independent vector will always form a plane rather than a Solid?
Regards, Tarun
2026-03-27 12:56:11.1774616171
Prove that linear combination of two 3D independent vector will form a plane rather than a Sold
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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.