Question 5
In the three-dimensional space with origin $O$, let $\mathbf i$, $\mathbf j$ and $\mathbf k$ be three mutually perpendicular unit vectors defined in the usual anticlockwise sense. Two intersecting lines $l$ and $m$ have equations $$l:\quad\mathbf r=\mathbf i+2\mathbf j+3\mathbf k+\lambda(\mathbf i+2\mathbf j-2\mathbf k),\quad\lambda\in\mathbb R,$$ and $$m:\quad\mathbf r=4\mathbf i+8\mathbf j-3\mathbf k+\mu(2\mathbf i-\mathbf j+2\mathbf k),\quad\mu\in\mathbb R,$$ respectively. The line with equation $$\mathbf r=\mathbf i+a\mathbf j+b\mathbf k+\omega(c\mathbf i+\mathbf j),\quad\omega\in\mathbb R$$ intersects the lines $l$ and $m$ and bisects the angle subtended by $l$ and $m$. Find the value of $|a+b+c|$.