[Math] ny difference between linear dependence, collinearity and coplanarity

linear algebravectors

The way it seems to me, linearly dependent vectors have to be collinear, and collinear vectors have to be coplanar. However, since a plane doesn't really have a direction, I'm assuming coplanar vectors can point in different directions as long as their lines exist on the same plane. Or do coplanar vectors/points also have to point in the same direction? If so, what's the practical difference between these concepts? I'm wondering this in terms of orientation and position in three-space, not in terms of whether the math is done differently or not.

For example, what would be the difference between 3 linearly dependent vectors, 3 collinear vectors and 3 coplanar vectors?

EDIT:
So far I still can't visualize the difference in 3D space. It's not that I don't understand that the math is different, I just want to be able to clearly visualize what the similarities and differences are, because if I don't then the math won't make sense to me. I need to understand what the math does in order to make it stick.

Best Answer

The way it seems to me, linearly dependent vectors have to be collinear, and collinear vectors have to be coplanar. However, since a plane doesn't really have a direction, I'm assuming coplanar vectors can point in different directions as long as their lines exist on the same plane.

Planes can certainly have a direction. In fact, planes have two directions, which can be specified by two vectors that span them. However, you are correct that coplanar vectors can point in different directions, so long as their lines are in the same plane. This is where the word coplanar comes from:

co-

together; mutually; jointly

planar

Of or pertaining to a plane.

So it means "together in a plane". Any group of vectors that are in the same plane are coplanar.

Or do coplanar vectors/points also have to point in the same direction?

No, as above, it only means in the same plane. If vectors are in the same plane and point in the same direction as you suggest, they are on the same line. Colinear of course means "together on a line". All colinear vectors happen to also be coplanar, but this is not embedded in the definitions of the words.

If so, what's the practical difference between these concepts? I'm wondering this in terms of orientation and position in three-space, not in terms of whether the math is done differently or not.

To address linear independence, I'll say the following.

  1. All colinear vectors are linearly dependent, almost trivially by the definitions of colinearity and linear dependence.

  2. For a set of non-zero coplanar vectors, none of which are colinear (i.e., they point in different directions), any two of the set can be considered linearly independent. This is because there are fundamentally only two orthogonal directions to "go" on a plane. As long as you have two vectors that are not colinear but lie in a plane, you can write any other member of the planar subspace as a linear combination of those two vectors.

Related Question