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