Q.By computing the shortest distance, determine whether following lines intersect each other. $\dfrac{x-5}{4} = \dfrac{y-7}{-5} = \dfrac{z+3}{-5}$ and $\dfrac{x-8}{7} = \dfrac{y-7}{1} = \dfrac{z-5}{3}$

[!TLDR] Apply the same intersection test to the Cartesian-form lines.

$\vec a_1=(5,7,-3)$, $\vec b_1=(4,-5,-5)$, $\vec a_2=(8,7,5)$, $\vec b_2=(7,1,3)$; the scalar triple product works out to $282\ne0$. …

Here the lines pass through $A(5,7,-3)$ with direction ratios $(4,-5,-5)$, and through $B(8,7,5)$ with direction ratios $(7,1,3)$.

Since this is not zero, the given lines do not intersect each other; and since $(4,-5,-5)$ is not proportional to $(7,1,3)$, the lines are not parallel either, so they are skew. …

Read off the point and direction ratios from each symmetric-form equation and apply the same intersection test as in part (i); a n …

Sign slips reading direction ratios off $\frac{y-7}{-5}$ and $\frac{z+3}{-5}$; expecting a 'clean' surd for the shortest distance -- textbook exercise (as opposed to worked-example) numbers do not a …