The Cartesian equation of line $L_1$ is $\frac{x-2}{a}=\frac{y+2}{b}=\frac{z-3}{c}$, where $a, b, c$ are constants. The line $L_2$ is parallel to the vector $4 \mathbf{i}+3 \mathbf{j}$. The line $L_3$ passes through the origin and the point with position vector $\mathbf{j}+\mathbf{k}$.
(i) Given that $L_1$ is perpendicular to $L_2$, form an equation relating $a$ and $b$.


(ii) Given that $L_1$ intersects $L_3$, show that $5 a+2 b-2 c=0$.


(iii) Hence express $a$ and $b$ in terms of $c$.


(iv) Find the acute angle between $L_1$ and $L_3$.