Let a line L passing through the point P(1,1,1) be perpendicular to...

Line L passes through P(1,1,1) perpendicular to two lines with directions $(4, 1, 1)$ and $(1, 1, 0)$.
Direction of L: $(4, 1, 1) \times (1, 1, 0) = (-1, 1, 3)$
Line L: $(x, y, z) = (1, 1, 1) + t(-1, 1, 3)$
Q is intersection with yz-plane (x=0): Setting $1-t=0$ gives $t=1$, so Q = (0, 2, 4)
Line through S(1,0,-1) parallel to L: $(x, y, z) = (1, 0, -1) + s(-1, 1, 3)$
R is intersection with yz-plane: Setting $1-s=0$ gives $s=1$, so R = (0, 1, 2)
For parallelogram PQRS: $\vec{PQ} = (-1, 1, 3)$ and $\vec{PS} = (0, -1, -2)$
Area = $|\vec{PQ} \times \vec{PS}| = |(1, -2, 1)| = \sqrt{1+4+1} = \sqrt{6}$
Square of area = 6