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Suppost $v$ be a vector and I want to project onto a non-orthonormal basis vector $u$. There is no span just two of these vectors.

How do I do that? Is it correct that if I say there is no span?

There are many question on this topic but I did not really understand it very well.

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jomegaA
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    If you have two vectors, then there also exists the span of those two vectors. But if you want the component of $v$ in the direction of $u$, you will need to know something about the rest of the basis. – David K Mar 03 '23 at 14:37
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    By "there is no span", you mean you want to project onto the subspace generated by $u$ alone, not one generated by some set of vectors, right? – eyeballfrog Mar 03 '23 at 14:38
  • @eyeballfrog yes the sub-space generated by $u$ alone. – jomegaA Mar 03 '23 at 14:42
  • @DavidK no no, not the component of the vector. There are basis vectors which represents the surface electric current density and there are basis vectors which represents the surface magnetic current density. I want to project basis vectors of surface magnetic current density on to surface electric current density. – jomegaA Mar 03 '23 at 14:45
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    In any inner product space, the projection of a vector $v$ onto a vector $u$ is given by $[(v\cdot u)/(u\cdot u)] u$. Intuitively, since the dot product is a measure of orthogonality, this is saying that the projection of $v$ onto $u$ gets smaller the closer $v$ is to orthogonal to $u$. Does that answer your question or is there something further you're confused about. – eyeballfrog Mar 03 '23 at 14:47
  • I got confused after reading this, https://math.stackexchange.com/questions/4311949/how-might-i-project-onto-a-non-orthogonal-basis/4312002#4312002 – jomegaA Mar 03 '23 at 14:48
  • @eyeballfrog In the above link, assume I just have a $u$ and $b$ – jomegaA Mar 03 '23 at 14:49
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    In the case of a single basis vector, the method used in that link will give the projection formula from my previous comment. You'll have gone through a bunch of extra steps, but the answer is the same. – eyeballfrog Mar 03 '23 at 14:54
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    If you are trying to do something like the linked question then you are trying to get the component of $v$ in the direction of $u.$ If you really need to do what that question is trying to do, with only $u$ and $v,$ you are out of luck. On the other hand, if you are not going to do anything with the other basis vectors after projecting $v$ onto $u,$ then the fact that $u$ came from a basis is irrelevant and the linked question is not what you want. I do not think we can advise you based on the information you have given; please state the problem much more explicitly. – David K Mar 03 '23 at 14:54
  • @DavidK "If you are trying to do something like the linked question then you are trying to get the component of $v$ in the direction of $u$" Yes this what I want. I will draw few pictures soon. – jomegaA Mar 03 '23 at 16:17
  • @DavidK, eyeballfrog I have added a picture. – jomegaA Mar 03 '23 at 16:36
  • If this is in two dimensions (as your diagram appears to indicate), $\rho_1,\rho_2,\rho_3$ are not a basis. They are a linearly dependent set of vectors. And I can't imagine how three vectors like that would relate to a current density (which you mentioned in an earlier comment). – David K Mar 03 '23 at 19:11
  • @DavidK Its in 3d. Actullay $\rho/2A$ model the current density in electric vector potential. I am kind of hobbyist so I did not dig much deeper into Linear Algebra or math in general. But very curious to learn. – jomegaA Mar 03 '23 at 19:19
  • Sorry, I can't make any sense out of what you've presented. This doesn't look like anything I encountered while studying electromagnetism. In order to make this a math question, you first have to understand the physics and be able to translate it correctly into math. If you have some source materials (such as a textbook) to refer to, maybe that can help, but I suspect that this is just going to be off-topic for this site. It might be on-topic for the physics stackexchange. – David K Mar 03 '23 at 19:42

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