With reference to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}=-\mathbf{i}+\mathbf{j}+2 \mathbf{k}$ and $\mathbf{b}=2 \mathbf{j}+5 \mathbf{k}$ respectively.

(i) Find a vector equation of the line $l_1$ that passes through point $A$ and is parallel to the vector $\mathbf{a}$.

[1]

(ii) Find the exact length of projection of $\mathbf{b}$ on $l_1$. Hence find $d$, the exact perpendicular distance from the point $B$ to $l_1$.

[4]

(iii) Using the value of $d$ found in part (ii), find the position vector of the point $C$, the foot of perpendicular from the point $B$ to $l_1$.

[3]

(iv) The line $l_2$ passes through point $B$ and is parallel to vector $\mathbf{b}$. Find a cartesian equation of $l_3$ which is the reflection of $l_2$ in $l_1$.

[3]