Show that vector $w=(3, 0, -1)$ is positioned in the plane that is set up by $u= (1, 2, 1)$ and $v= (2, 1, 0)$. Provide the coordinates for $w$ expressed in the base $\{u,v\}$ as well.
How do I answer the question? I know that if I use the linear combination of the vectors I can show that vectors $w,u,v$ are complanar. However, for the second question, I also need to provide coordinates for $w$ in that plane, which I don't even know how to begin with.
If you could write $w = 17u + (-3)v$, for instance, then the answer to the second part would be "in the $u,v$-coordinate system, the coordinates of $w$ are $17$ and $-3$."
So if you manage to answer the first part (using "I know that if I use the linear combination of the vectors I can show that vector $$ is indeed in a plane of $$ and $$") then you'll have solved the second part as well.