How many planes can two three-dimensional vectors fill in the three-dimensional space?

Question

In the "Introduction to linear algebra", G. Strang gives an example of two 3d vectors (say v=[1,0,0] and w= [0,2,3] ) and says that

combinations of cv + dw fill a plane in R³.

I read that the 3d space has multiple planes. Hence, the question: can it be only one plane that those combinations fill or is it possible that those linear combinations of two 3D vectors fill multiple planes in 3D?

Answer 1

A plane in $\mathbb{R}^3$ is defined by a $2$D direction (subspace of $2$ linearly independent vectors, i'll refer to them as $u,v\in\mathbb{R}^3$ ) and a point (i'll refer to it as $P\in\mathbb{R}^3$). That subspace $\text{span}(u,v)$ defines an infinite number of planes in $\mathbb{R}$, depending on the $P$ point.

Despite this, any 2 planes defined by the subspace $\text{span}(u,v)$ will be parallel between them, so if you're just considering $P$ as $O$ (coordinates origin, $(0,0)$), they just define one plane.

Answer 2

Here is one way to think about it: If you have two vectors $v$ and $w$, imagine adding $v$ to the head of $w$, and $w$ to the head of $v$. Then you get a parallelogram in 3D whose vertices are: the origin, $v$, $w$, and $v + w$. This is the special case $c = 1, d = 1$. Now can you see how to generalize this to all possible linear combinations $cv + dw$? Do you see how we fill up the plane that contains that parallelogram (and for a given parallelogram, there is only one plane containing it)?