Is there a hard proof why 2 non-parallel, non-orthogonal vectors can always define an entire plane?

Question

Whenever I try to visualize making up all points in a plane using 2 non-parallel vectors I fail to find any points that I can't make up by a linear combination of these vectors. However, I'm very interested in knowing a proof of such a powerful statement and was wondering if there is one. I understand that the case where these vectors are perpendicular doesn't require a proof.

Answer 1

The key point is that two not parallel vectors are linearly independent and therefore their span is a plane.

Note that we can always reduce to the orthogonal case indeed given vectors v and w not parallel and not orthogonal let consider

$$u=v-\frac{v^Tw}{w^Tw}w$$

and since u is orthogonal to w they span a plane which is the same spanned by v and w since u is a linear combination of them.

Answer 2

Hint: Let $\vec{u},\vec{v}$ be two non-parallel vectors. If $\vec{i},\vec{j}$ are the usual orthogonal vectors of length $1$, you can write: $$\vec{u}=u_x·\vec{i}+u_y\cdot \vec{j},\ \vec{v}=v_x·\vec{i}+v_y\cdot \vec{j}$$ Can you proceed from that to show that any $\vec{x}$ on the plane can be written as $x_u\vec{u}+x_v\vec{v}$?