Answer
$\textbf{a}=⟨2,0⟩+⟨0,-3⟩$
Work Step by Step
We have the vectors $a=<2,-3>$ and $b=<5,0>$. Then:
$\textbf{a}_{\parallel \textbf{b}}=(\frac{\textbf{a}\cdot\textbf{b}}{\textbf{b}\cdot\textbf{b}})\textbf{b}=\frac{2\times5+(-3)\times0}{5^{2}+0^{2}}⟨5,0⟩=⟨2,0⟩$
$\textbf{a}_{\perp \textbf{b}}=\textbf{a}-\textbf{a}_{\parallel \textbf{b}}=⟨2,-3⟩-⟨2,0⟩$
$=⟨0,-3⟩$
Therefore, the decomposition of $\textbf{a}$ with respect to $\textbf{b}$ is
$\textbf{a}=\textbf{a}_{\parallel \textbf{b}}+\textbf{a}_{\parallel \textbf{b}}=⟨2,0⟩+⟨0,-3⟩$
