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?
Best Answer
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.