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Adding a third vector that pokes out of the plane at an angle allows the entire 2D sheet to slide up and down through space. With three freely varying scalars ($a\vec{v} + b\vec{w} + c\vec{u}$), the span expands from a plane to encompass every conceivable point in the 3D volume, effectively filling all of 3D space.

Conditions: Starting with two linearly independent vectors spanning a plane; Introducing a third vector that is NOT in that plane

Adding a third vector that pokes out of the plane at an angle allows the entire 2D sheet to slide up and down through space. With three freely varying scalars ($a\vec{v} + b\vec{w} + c\vec{u}$), the span expands from a plane to encompass every conceivable point in the 3D volume, effectively filling all of 3D space.

Conditions: Starting with two linearly independent vectors spanning a plane; Introducing a third vector that is NOT in that plane