Given u =[1,-1,0], v=[0,1,-1] and w=[-1,0,1], I realize that these are not independent vectors since $-1\cdot u - v = w$ , but I struggle to "see" how they are on the same plane --and i've found an app to allow me to easily plot them in 3D I can see it, but I still can't.
I know this is basic, but I'm still struggling to see this. Thank you.
Vector $w$ is in a plane generated by two nonparallel vectors $u$ and $v$ if we can express $w$ with $u$ and $v$. So there must be some scalars $a,b$ such that:
$$ w=au+bv$$
Now it is easy to see that $a=b=-1$ satisfies this equation.