I've read something about how collinear vectors don't "establish a plane", but as far as I'm concerned they would still exist along one single plane when their directions are all the same. Or does collinearity not mean existing along the same line? Or does existing along the same line somehow not imply existing on the same plane?
[Math] Are 3 collinear vectors also coplanar
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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.
All colinear vectors are linearly dependent, almost trivially by the definitions of colinearity and linear dependence.
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.
$a\cdot(b \times c)=(a \times b) \cdot c=0$
This implies $c$ is perpendicular to $a \times b$.
So what's the relation with $c$ and $a,b$ ? (Note: $a \times b$ is perpendicular to both $a$ and $b$).
Do you observe that c is parallel to the plane spanned by a and b?
therefore c= xa+yb for some x,y in R.
Best Answer
Think about the following arbitrary case in $\mathbb R^3$: