consider points p(2,1,3), q(1,2,1), R(-1,-1,-2), s(1,-4,0)
Find the shortest distance between lines pq and rs
Find the constant a so that the following vectors are coplanar
2i-j+k i+2j-3k 3i-j+2k
Is it clear to you these two lines are skew lines?
The distance between two skew lines, $\displaystyle P + t\overrightarrow d \,\& \,R + s\overrightarrow e $ then the distance between them is:
$\displaystyle \frac{{\left| {\overrightarrow {PR} \cdot \left( {\overrightarrow d \times \overrightarrow e } \right)} \right|}} {{\left\| {\overrightarrow d \times \overrightarrow e } \right\|}}$
Magnitude. Of course I'm also assuming he means the same with the single bars in the top. It essentially is the exact same thing as an "absolute value" sign.
As for your second question, for three vectors to be coplanar, their scalar triple product must be 0. My problem, is I see no constant "a" in any of those vectors that one would have to solve for. Is their information missing from the problem.
The reason for the single bars on top is that the dot product is a number
hence we have the absolute value of a number.
On the bottom we have double bars because the cross product is a vector so we can talk about the magnitude of a vector but not the absolute value of a vector.