I think there may be an issue with this question.
Here, you see that line L1 can be seen as parallel to the following vector based on the points it passes through
`vec(L1) = <P_(x), P_(y), P_(z)> - <Q_(x), Q_(y), Q_(z)>`
`vec(L1) = <-2, -1, 4>`
Suppose now that we have a vector that represents the direction of L2, that is parallel to <3, 2, 4>. This would indicate that:
`vec(L2) = c<3,2,4>`
where "c" is a constant multiplied by the vector.
Here's where the problem breaks down. In order for L2 to be perpendicular to L1, `vec(L1)*vec(L2) = 0`
However, the dot product is not zero as we see here:
`vec(L1)*vec(L2) = (-2*3c + -1*2c + 4*4c)`
`=c(-6-2+16) = 8c`
The only way 8c is zero is if `vec(L2) = vec(0)`, which is a trivial case.
Therefore, no line can be found that is both perpendicular to L1 and that is parallel with the vector <3, 2, 4>.
Also, "p" (lower-case p) hasn't been defined in the problem, so we can't tell where it would be an integer.
Of course, I'm assuming that this is a straight line. If it's not, this becomes a whole different animal...
Please correct me, by the way, if I'm interpretting something the wrong way!