Orthogonality & Least Squares · Hard problem

A light ray travels along direction $\mathbf d = (1,2,2)$ and strikes a planar mirror with normal $\mathbf n = (2,-1,2)$. The reflected ray's direction is $\mathbf d' = \mathbf d - 2\left(\dfrac{\mathbf d\cdot\mathbf n}{\mathbf n\cdot\mathbf n}\right)\mathbf n$. Find the first coordinate of $\mathbf d'$.