What does it mean when a set of vectors v1, v2, …, vn , is linearly dependent if a linear combination satisfy the following condition: a1v1 + a2v2 + … + anvn = 0, for some scalar a, not all of them equal to 0 ?
Is it because if these vectors are on the same plane, vectors that are non-zero can add up to cancel out? Thats why coplanar vectors are linearly dependent? while vectors that point out of the plane cannot be cancelled out, so it is linearly independent?
When you've got two linearly independent vectors (in $\mathbb{R}^n$) they span a plane, as linear independence in the case of two vectors means that they are not parallel. Now all linear combinations of these two vectors will exactly be all of the points on the plane, meaning any vector linearly independent of those two would not be in the plane they span (just called the "span" of these vectors).
In general a vector is called linearly independent of a set of vectors, if it can not be expressed as a linear combination of the latter. This just amounts to the above in the case of two vectors spanning a plane.