. Background Information:

I am studying linear algebra regarding vectors, planes, and lines.

. Logical Question:

How would you know if some vectors lie on a plane, or a line, or both?

. My Solution:

1. Vectors on the same plane:

Few vectors are on the same plane if they are linearly dependent, by testing Av1 + Bv2 + Cv3 = 0 or Av1 = -Bv2 - Cv3 (subtration of two other vectors yields one of the vectors); considering A, B, and C are not 0. Therefore, if the vectors are linearly dependent then they are on the same plane.

2. Vectors on the same line:

Few vectors are considered to be on the same line if they are scalar multiples of of one another. For example, v1 = 2v2 , v3 = 3v2.

Am I right? If you could confirm this for me it would be great.

  • $\begingroup$ By plane do you mean that they comprise a subspace of dimension 2? $\endgroup$ – eepperly16 Dec 15 '17 at 0:36
  • $\begingroup$ Thanks for reaching out to help, by plane I am talking about 3 dimension. $\endgroup$ – Kourosh Dec 15 '17 at 0:38
  • $\begingroup$ Please use MathJax to format your posts. $\endgroup$ – Chase Ryan Taylor Dec 18 '17 at 1:48
  • $\begingroup$ Sure I'll look into that. Thanks! $\endgroup$ – Kourosh Dec 21 '17 at 4:15

I'm going to presume from the nature of your question that you're talking about vectors in $\mathbb{R}^3$.

You're sort of right. Certainly if $x$ and $y$ are in the same one dimensional subspace of $\mathbb{R}^3$ (i.e. lie on a line) then we can say $x = \lambda y$ for some $\lambda \in \mathbb{R}^3$.

If you have three vectors and you want to see that they lie in a plane, you can indeed check whether they are linearly independent, as you suggest. If they are, then they span a 3 dimensional subspace. If only two of them are linearly independent, then they span a 2 dimensional subspace.

  • $\begingroup$ Yes I am working with vectors in R^3, sorry I forgot to mention it. Thanks for your clarification. I will accept the answer in a few min. $\endgroup$ – Kourosh Dec 15 '17 at 0:42
  • $\begingroup$ I forgot to ask, can the vectors occur on both the line and the plane? I believe yes. $\endgroup$ – Kourosh Dec 15 '17 at 0:44
  • $\begingroup$ I don't know what you mean by that question $\endgroup$ – Matt Dec 15 '17 at 0:46
  • $\begingroup$ I mean would it be possible for few vectors to satisfy x=λyx=λy for some λ∈R3 and Av1 + Bv2 + Cv3 = 0? In my book it does not mention anything about this case, but I wanted to know out of curiosity. $\endgroup$ – Kourosh Dec 15 '17 at 0:55
  • $\begingroup$ If you're asking whether three vectors can lie on the same line and also lie in the same place, yes they can, because the line they're laying on lies in a plane. $\endgroup$ – Matt Dec 15 '17 at 0:57

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