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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 :)

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just for start, what have you tried so far? – V-X Mar 18 '13 at 6:58
@George Randall Why do you want this post to be deleted? – Eric Naslund Mar 19 '13 at 1:23
up vote 1 down vote accepted

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 $$\frac{|\bf u\times v|}{|\bf u|}$$

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and apparently there's like 2 different methods, one using the cross product (which jared has clearly shown) and one using the dot product. – George Randall Mar 18 '13 at 8:32

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