Vectors on a Plane vs on a Line - Linear Algebra

. 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?

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.

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