Answer
$\textbf{a}=⟨0,-\frac{1}{2},-\frac{1}{2}⟩+⟨4,-\frac{1}{2},\frac{1}{2}⟩$
Work Step by Step
We are given the vectors $a=<4,-1,0>$ and $b=<0,1,1>$. We have:
$\textbf{a}_{\parallel \textbf{b}}=(\frac{\textbf{a}\cdot\textbf{b}}{\textbf{b}\cdot\textbf{b}})\textbf{b}=\frac{4\times0+(-1)\times1+0\times1}{0^{2}+1^{2}+1^{2}}⟨0,1,1⟩=⟨0,-\frac{1}{2},-\frac{1}{2}⟩$
$\textbf{a}_{\perp \textbf{b}}=\textbf{a}-\textbf{a}_{\parallel \textbf{b}}=⟨4,-1,0⟩-⟨0,-\frac{1}{2},-\frac{1}{2}⟩$
$=⟨4,-\frac{1}{2},\frac{1}{2}⟩$
Therefore, the decomposition of $\textbf{a}$ with respect to $\textbf{b}$ is
$\textbf{a}=\textbf{a}_{\parallel \textbf{b}}+\textbf{a}_{\parallel \textbf{b}}$
$=⟨0,-\frac{1}{2},-\frac{1}{2}⟩+⟨4,-\frac{1}{2},\frac{1}{2}⟩$
