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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.

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  • $\begingroup$ 3 linearly dependant vectors are coplanar, it is necessary and sufficient (and easy to see, from a relation $\lambda_1v_1 + \lambda_2v_2 + \lambda_3v_3=0$). However, 3 colinear vectors span a dimension 1 subspace, whereas coplanar vectors can span a dimension 2 subspace (it can be 1 if the vectors are actually colinear). This is the only difference. $\endgroup$
    – zarathustra
    Jan 7, 2014 at 16:24
  • $\begingroup$ Linearly dependent vectors need not be co-linear. For example you can take the coordinate axis in $\mathbb{R}^2$ and the vector $\mathbf{v}=\begin{pmatrix} 1 \\ 1 \end{pmatrix}$. Clearly $\{\mathbf{v},\mathbf{e}_1,\mathbf{e}_2\}$ is a set of linearly dependent vectors but none of them are co-linear. $\endgroup$
    – Ben
    Jan 7, 2014 at 16:27

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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.

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A set $S$ of vectors is called collinear iff $\dim\bigl(\mathrm{span}(S)\bigr)=1$ and iff that dimension is $2$ the vectors are called complanar.

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Any two vectors (which are NOT linearly dependent, i.e. not a scaling of each other) establish a plane. Given another vector which is in the span of these vectors, it is "coplanar" with them (in the same plane). So being coplanar does mean linear dependence (to the basis of a given plane). Colinear is the same idea but more general, the dependence doesn't have to be in a plane, it can be a hyperplane etc.

As a motivation, if we take the standard basis $E=\{e_1,e_2,e_3\}$ and the vector $v=(1,1,1)$, $v$ is linearly dependent with the elements of $E$, but not coplanar with any pair of $\{e_i , e_j \}$. However, $w=(2,4,0)$ is coplanar with $\{e_1,e_2\}$ (these examples are easier to see because of use of standard basis). By the way, Arnold's Classical Mechanics does make appeal to these ideas quite frequently.

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