Answer
$\textbf{a}=⟨4,2,2⟩+⟨0,-3,3⟩$
Work Step by Step
We are given the vectors $a=<4,-1,5>$ and $b=<2,1,1>$. We have:
$\textbf{a}_{\parallel \textbf{b}}=(\frac{\textbf{a}\cdot\textbf{b}}{\textbf{b}\cdot\textbf{b}})\textbf{b}=\frac{4\times2+(-1)\times1+5\times1}{2^{2}+1^{2}+1^{2}}⟨2,1,1⟩=⟨4,2,2⟩$
$\textbf{a}_{\perp \textbf{b}}=\textbf{a}-\textbf{a}_{\parallel \textbf{b}}=⟨4,-1,5⟩-⟨4,2,2⟩$
$=⟨0,-3,3⟩$
Therefore, the decomposition of $\textbf{a}$ with respect to $\textbf{b}$ is
$\textbf{a}=\textbf{a}_{\parallel \textbf{b}}+\textbf{a}_{\perp \textbf{b}}$
$=⟨4,2,2⟩+⟨0,-3,3⟩$
