Question

Can we always find a plain that contains any two vectors in space? So this is my third time asking in Chegg, one expert answered: "Yes, just as we can connect any two points, we can find a plane that contains any two lines/vectors". The other expert answered: "No, if they are subspace of different vector spaces, we cannot find a plane the two vectors lies in".
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I am confused. I tried picturing finding a plane for some vectors in space, and it sometimes seems impossible. Please explain this (preferably graphically, but any clarification will do)

Thanks in advance

Answer 1

The answer is yes.

Given two vectors in a finite dimensional vector space, there are two cases you can consider.

1. The two vectors are linearly dependent

2. The two are linearly independent

In case two, if you consider the span of the two vectors, you get a plane containing the two vectors. In case one, the span of the two vectors only gives you a line. All you need to do here is find another vector which is linearly independent to the given vectors and consider the span of these three vectors. Note here that for this to be possible, your vector space should be of dimension at least two. Otherwise the entire space itself is a line (when dimension is 1) or it is single point ( when the dimension is 0 in which case the vector space is {0}).