4. Let $v = (0, 1, 1)$ and $U = \{u \in \mathbb{R}^3 \mid \text{proj}_v u = 0\}$
a) Show that if $u = (x, y, z) \in \mathbb{R}^3$, then $\text{proj}_v u = \frac{y + z}{2}(0, 1, 1)$
b) Find a Cartesian equation for $U$, i.e., find $a, b, c, d \in \mathbb{R}$ such that
$U = \{(x, y, z) \in \mathbb{R}^3 \mid ax + by + cz = d\}$,
Give a complete geometric description of $U$.
c) Is $U$ a subspace of $\mathbb{R}^3$?
d) Find a spanning set for $U$.