Dimension of a plane in $\mathbb R^3$ through the origin

Question

If I have a plane in $\mathbb R^3$ like this $x+y+z=0$, can I say the dimension is $2$ because it is spanned by $2$ independent vectors and it goes through the origin? I am a bit confused because in the parametric equation I need a position vector and $2$ other vectors and therefore the plane is spanned by $3$ linear independent vectors and has dimension $3$, but that's impossible because it does not span the whole $\mathbb R^3$ . So where did I go wrong?

Thank you very much for your help!

Answer 1

In the parametric plane equation $r=r_0+s\vec{v}+t\vec{w}$, $\vec{v}$ and $\vec{w}$ are linearly independent vectors and $r_0$ is the vector representing the position of an arbitrary (but fixed) point on the plane. So you dont need 3 vectors, you need one point and two vectors linearly independent. Therefore the plane has dimension 2