I'm struggling with the general equation of this set of vectors algebraically.

Question

I'm struggling with Part B in the problem below. I thought the answer would be x(-1, 0, 1) + y(1, -1, 0), but that doesn't seem to be correct. Any assistance would be greatly appreciated!

Answer 1

$u$ and $v$ are linearly independent, but $w=-u-v$, so the span of these $3$ vectors is a ($2$-dimensional) plane through the origin; that is, the coordinates $(x,y,z)$ of the space spanned by these vectors satisfy $Ax+By+Cz=0$ for some $A, B, $ and $C$.

Since $u$ and $v$ are in the plane, we know that $A\times-1+C\times1=0$ and $A\times1+B\times-1=0$; i.e., $A=C$ and $A=B$. Thus, the plane equation is $Ax+Ay+Az=0$, or $x+y+z=0$.