# When a third vector in a plane does not lie in the span of 2 linearly independent vectors in the plane

For instance, can the 3 vectors $$\vec a=[1, \ 0, \ 1]^T, \vec b=[2, \ 7, \ -2]^T, \vec c=[3, \ 1,\ 5]^T$$ lie on the same plane in $$\mathbb R^3$$?

My understanding is that the span of 2 linearly independent vectors in $$\mathbb R^3$$ such as $$\vec a$$ and $$\vec b$$ is a plane that passes through $$\vec 0$$. Yet $$\vec c$$ is not in the span of $$\vec a$$ and $$\vec b$$. What's happening?

I can see $$a^Tb=0$$. Is that relevant? According to Wikipedia, "Note that v and w can be perpendicular, but cannot be parallel." I take to understand that $$\vec c$$ would be the $$\vec r$$ and $$\vec 0$$ would be the $$\vec r_0$$ in the equation that was given $$\vec r-\vec r_0=s\vec v+t\vec w$$

(This is an edit to add a new idea): Do the vectors span $$\mathbb R^3$$ because they are all linearly independent? Then it that case, what does this mean? What's the plane that is spanned by 3 linearly independent vectors?

• I assume you mean $[1,0,1]$, not $$, and $[2,7,-2]$, not $[27-2]=$. – mr_e_man Oct 6 '18 at 23:57
• @mr_e_man Yes. I thought spacing would helping. Anyway, I put commas. Thanks. – user198044 Oct 6 '18 at 23:57
• I think you should look at the distinction between linear subspaces and affine subspaces. A plane is determined by 3 points. The 3 points $\vec0,\vec a,\vec b$ determine a plane, which is a linear subspace. The 3 points $\vec a,\vec b,\vec c$ determine a plane, which is an affine subspace. The 4 points $\vec0,\vec a,\vec b,\vec c$ determine the entire 3D space. – mr_e_man Oct 7 '18 at 0:11
• @mr_e_man Can you help me here? How is a plane a particular solution plus the span of linearly independent vectors? – user198044 Oct 7 '18 at 0:12
• An affine subspace is a linear subspace displaced from the origin. The "particular solution" is that displacement. – mr_e_man Oct 7 '18 at 0:25

There is nothing wrong with a vector in $$\mathbb R^3$$ not being in the span of two linearly independent vectors, since $$\mathbb R^3$$ is $$3$$ dimensional, whereas the span of the two vectors will be only $$2$$ dimensional, i.e. a plane.
• Thank you! I just figured out that the vectors span $\mathbb R^3$. So what is the "plane" formed by 3 vectors than span $\mathbb R^3$? – user198044 Oct 7 '18 at 0:06
• $\mathbb R^3$ itself; though it is not usually called a "plane". – Chris Custer Oct 7 '18 at 0:08