Let S be the subspace of R4 containing all vectors with x1 + x2 + x2 + x4 = 0. Find a basis for the space S orthogonal, containing all vectors orthogonal to S.
There are a few ways to solve this but first note that basis for S is
$\displaystyle (x_1,x_2,x_3,-x_1-x_2-x_3)=x_1(1,0,0,-1)+x_2(0,1,0,-1)+x_3(0,0,1,-1)$
So you could just find a vector perpendicular to these three above vectors.
Or if you take the gradient of $\displaystyle x_1+x_2+x_3+x_4=0$ you get
$\displaystyle (1,1,1,1)$ you can check using the vectors above that this is orthogonal to the space S.