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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.

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.

The span of two arbitrary vectors $\vec{v}$ and $\vec{w}$ that do not lie on the same line is the entire 2D plane. This is achieved by taking all possible linear combinations $a\vec{v} + b\vec{w}$ where scalars $a$ and $b$ vary continuously across all real numbers.

Conditions: Two vectors $\vec{v}$ and $\vec{w}$ are nonzero; Vectors $\vec{v}$ and $\vec{w}$ are non-collinear (point in different directions)

The span of two arbitrary vectors $\vec{v}$ and $\vec{w}$ that do not lie on the same line is the entire 2D plane. This is achieved by taking all possible linear combinations $a\vec{v} + b\vec{w}$ where scalars $a$ and $b$ vary continuously across all real numbers.

Conditions: Two vectors $\vec{v}$ and $\vec{w}$ are nonzero; Vectors $\vec{v}$ and $\vec{w}$ are non-collinear (point in different directions)