coplanar vectors

1MileCrash

Two free vectors are always coplanar.

Then if A and B are free vectors, are A, B, and A+B all coplanar in all cases?

Mark44

Yes.
Any two vectors that start from the same point (you can assume that they start from the origin) determine a plane. Any linear combination of these vectors (including 1*A + 1*B) also lies in that same plane.

1MileCrash

Cool.

How about this:

If two vectors are linearly dependent, they are collinear. They are always coplanar.

If three vectors are linearly dependent, they are coplanar. Three vectors are always all co"cubeular" (I don't know a word like coplanar for a three dimensional object.)

Based on this pattern, it correct to say that:

If n vectors are linearly dependent, then they are co(n-1 space object) and are always co(n space object).

Since for two vectors, an n-1 space object is a line, for three it is a plane, and so on.

Am I making sense?

Mark44

Yes. If two vectors are linearly dependent, then each is a nonzero scalar multiple of the other.
They could be collinear, depending on which vectors we're talking about.
I don't believe there is any special terminalogy beyond coplanar.
Yes, I get what you're saying, but as I said, I don't believe there is any terminology beyond coplanar.