Coplanar Vectors: A, B, & A+B

[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]

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.
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]

Cool.

How about this:

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

Yes. If two vectors are linearly dependent, then each is a nonzero scalar multiple of the other.

If three vectors are linearly dependent, they are coplanar.

They could be collinear, depending on which vectors we're talking about.

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

I don't believe there is any special terminalogy beyond coplanar.

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?

Yes, I get what you're saying, but as I said, I don't believe there is any terminology beyond coplanar.

 

[CellCoree]

Yes, A, B, and A+B will always be coplanar in all cases. This is because coplanar vectors are defined as vectors that lie in the same plane, and since A and B are free vectors, they can be translated and rotated without changing their magnitude or direction. Therefore, when A and B are added, the resulting vector A+B will still lie in the same plane as A and B, making all three vectors coplanar. This is a fundamental property of vector addition and applies to all cases.