Two free vectors A and B are always coplanar, as they define a plane when originating from the same point. Any linear combination of these vectors, including A + B, also lies within that plane. If two vectors are linearly dependent, they are collinear and thus coplanar. Extending this concept, if three vectors are linearly dependent, they remain coplanar, and this pattern holds for n vectors, where linearly dependent vectors are co(n-1 space object) and co(n space object).
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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?