Independence

You have seen redundancy twice now: $(2,4)$ added nothing to $(1,2)$. Time to make that precise, because "points in a different direction" is not yet a test you can apply.

The definition, built from the failure

Vectors are linearly dependent when one of them can be written as a linear combination of the others. It sits inside the span of the rest, so it extends nothing.

$(1,2)$ and $(2,4)$: dependent, since $(2,4) = 2(1,2)$.

$(1,2)$ and $(0,1)$: independent. No scalar multiple of $(1,2)$ produces $(0,1)$, because that would require the first coordinate $1 \cdot c = 0$, forcing $c = 0$, which gives $(0,0)$ instead.

Rotate the second vector in the visual. Almost always the pair spans the plane; at the exact instant it aligns with the first, the span snaps down to a line.

Three vectors in a plane

Take $(1,0)$, $(0,1)$, and $(3,5)$. The first two already reach everywhere, so the third is reachable too: $(3,5) = 3(1,0) + 5(0,1)$. Dependent.

This is forced, not bad luck. In a plane, any three vectors are dependent. Two independent directions already exhaust the plane, so a third has nowhere new to point. In 3D space, any four vectors are dependent, for the same reason.

Why this is the pivot of Act I

Independence is what makes a basis a basis: a basis is exactly a set that is independent and spans.

It also decides the size of things. A plane needs 2 independent vectors, no more and no fewer. That count is forced by the space itself, not by your choice of basis — and a forced number is worth naming.

Next: naming it, and finding the flat worlds hiding inside bigger spaces.