Vector / geometry question 
I've spent the last 2 hours trying to solve this question, but it's just too hard. Could someone please explain to me in a step by step manner on how I would go about this question.
Help would be very much appreciated :)

First, notice that ${\bf d_1\times d_2}={\bf N}$ represents a vector perpendicular to both $\bf d_1$ and $\bf d_2$.  The perpendicular distance between the two lines will thus be the length of the vector projection of ${\bf v}=(x_1-x_2,y_1-y_2,z_1-z_2)$ onto $\bf N$.  This is exactly given by the formula
$$\frac{\bf v\cdot N}{|\bf N|}$$
For part $(b)$, the hint is very helpful.  The distance between the two parallel lines will be the height of the parallelogram defined by the vectors $\bf v$ (defined above) and ${\bf u}=(a_1,b_1,c_1)$.  Draw a quick picture to see this fact.  Recalling that the area of said parallelogram is $|\bf u\times v|$, we see that the distance between the two lines is given by
$$\frac{|\bf u\times v|}{|\bf u|}$$
