The discussion centers on the coplanarity of vectors, specifically whether two free vectors A and B, along with their sum A+B, are always coplanar. Participants explore the implications of linear dependence among vectors and the associated geometric interpretations.
Participants generally agree on the coplanarity of two vectors and their sum, but there is some disagreement regarding the terminology and the implications of linear dependence for three or more vectors. The discussion remains unresolved regarding the generalization of coplanarity for n vectors.
Participants express uncertainty about the terminology for higher-dimensional coplanarity and the implications of linear dependence, which may depend on specific definitions and assumptions about vector spaces.
Readers interested in vector mathematics, linear algebra, and geometric interpretations of vector relationships may find this discussion relevant.
Yes.1MileCrash said: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?
Yes. If two vectors are linearly dependent, then each is a nonzero scalar multiple of the other.1MileCrash said:Cool.
How about this:
If two vectors are linearly dependent, they are collinear. They are always coplanar.
They could be collinear, depending on which vectors we're talking about.1MileCrash said:If three vectors are linearly dependent, they are coplanar.
I don't believe there is any special terminalogy beyond coplanar.1MileCrash said:Three vectors are always all co"cubeular" (I don't know a word like coplanar for a three dimensional object.)
Yes, I get what you're saying, but as I said, I don't believe there is any terminology beyond coplanar.1MileCrash said: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?