Geometrical explanation / visualisation as to why 2 vectors cannot span R3?

Question

I have already understood as to why 2 vectors cannot span R3 in a simple algebraic manner. This is from an answer to this very question 3 years ago : Why two vectors cannot span ${\bf R}^3$?

However, I would like to visualise as to why it is so. I have tried visualizing in my mind as well as drawing by hand in whatever way possible, but still can't wrap my head around it.

Answer 1

If you are drawing two vectors, you are probably doing it on a planar object, possibly a sheet of paper. Now, use an appropriate parallelogram, similar to the one they teach you when you learn how to add vectors, to draw any linear combination of these two vectors (assuming they are not collinear, which if they are - they certainly span just a line). Notice that all such parallelograms remain in the same plane where you had drawn the two vectors. Conclusion: Every pair of (linearly independent) vectors span a plane.

Answer 2

Hint

Any vector that lies in the linear subspace spanned by two vectors lies in a plane (the one spanned by the two vectors). You get the desired result as $\mathbb R^3$ is not included in a plane.