Step 1 :Calculate the dot products and the norms: \(\mathbf{y} \cdot \mathbf{u}_{1} = 17*1 + 1*0 + 5*(-1) = 12\), \(\mathbf{y} \cdot \mathbf{u}_{2} = 17*2 + 1*1 + 5*2 = 39\), \(\left\| \mathbf{u}_{1} \right\|^2 = 1^2 + 0^2 + (-1)^2 = 2\), \(\left\| \mathbf{u}_{2} \right\|^2 = 2^2 + 1^2 + 2^2 = 9\)
Step 2 :Substitute these values into the formula: \(\operatorname{proj}_{W} \mathbf{y} = \frac{12}{2} \mathbf{u}_{1} + \frac{39}{9} \mathbf{u}_{2} = 6\mathbf{u}_{1} + \frac{13}{3}\mathbf{u}_{2}\)
Step 3 :Calculate the projection: \(\operatorname{proj}_{W} \mathbf{y} = 6\left[\begin{array}{r}1 \\ 0 \\ -1\end{array}\right] + \frac{13}{3}\left[\begin{array}{l}2 \\ 1 \\ 2\end{array}\right] = \left[\begin{array}{r}16 \\ \frac{13}{3} \\ 4\end{array}\right]\)
Step 4 :So, the closest point to \(\mathbf{y}\) in the subspace \(W\) is \(\boxed{\text{B. }\left[\begin{array}{r}16 \\ 5 \\ 4\end{array}\right]}\)