How does adding a third vector outside the plane of two others affect the span in 3D space?

A third vector lying perfectly flat on the plane created by the first two provides no new directional freedom. Mathematically, it is redundant because it can be written as a linear combination of the other two vectors.

Conditions: Three vectors exist in 3D space; The first two vectors are linearly independent (non-parallel); The third vector lies within the plane spanned by the first two

To form a basis, a set of vectors must be both linearly independent and span the target space. Linear independence means each vector contributes a brand-new dimension that cannot be recreated by the others.

Conditions: The vectors belong to the target vector space; The set is minimal (no redundant vectors)

A linear combination of two arbitrary non-collinear vectors $\vec{v}$ and $\vec{w}$ is formed by multiplying each vector by a scalar ($a$ and $b$) and adding them together ($a\vec{v} + b\vec{w}$). If both scalars are allowed to vary continuously across all real numbers, the tips of these combined vectors sweep out every possible point on the 2D plane.

Conditions: The two vectors $\vec{v}$ and $\vec{w}$ are non-collinear (point in different directions and do not lie on the same line).; The scalars $a$ and $b$ vary continuously across all real numbers.

In 3D space, the span of two non-parallel vectors forms a tilted, infinitely extending flat sheet cutting through the origin. While each individual vector spans a line, their linear combination $a\vec{v} + b\vec{w}$ traces out this entire 2D plane embedded within the 3D volume.

Conditions: Working in three-dimensional Euclidean space; Two vectors $\vec{v}$ and $\vec{w}$ are non-parallel

In 3D space, the span of two non-parallel vectors is a tilted, infinitely extending flat sheet cutting through the origin. While their individual spans form lines, combining them via linear combinations ($a\vec{v} + b\vec{w}$) traces out this exact 2D plane.

Conditions: The vectors are in three-dimensional space.; The two vectors are non-parallel.