Final answer:
The requested vectors are v1 = (1, 0), v2 = (0, 1), and v3 = (1, 1). They are co-planar in the xy-plane, with v1 and v2 as independent vectors, while v3 is dependent on v1 and v2.
Step-by-step explanation:
The question asks us to find three vectors v1, v2, and v3 such that any two of these non-zero vectors span a plane and the span of all three vectors is also a plane.
To achieve this, all three vectors must be co-planar but no vector can be a scalar multiple of another vector, ensuring that any two vectors span a plane. Since span {v1, v2, v3} is also a plane, it means the third vector must be linearly dependent on the first two vectors.
Next, we need to find a third vector, v3, that lies in the same plane spanned by v1 and v2. One way to do this is to take the cross product of v1 and v2. The cross product of two vectors in three dimensions gives us a vector that is perpendicular to both of the original vectors and therefore lies in the same plane as them.
An example of such vectors in a two-dimensional xy-plane could be v1 = (1, 0), v2 = (0, 1), and v3 = (1, 1). Here, v1 and v2 are the standard basis vectors, and v3 is the sum of v1 and v2. Any two vectors here span a plane because they are linearly independent, and all three vectors also span a plane as v3 is linearly dependent on v1 and v2.