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The variable line $l_1$ passes through the point $A$ $(2, 1, -1)$ and is parallel to the direction $ti + j +(1-t)k$

The variable line $l_2$ passes through the point B(1, -2, 3) and is parallel to the direction $-2i +tj +(t+2)k$

The points $P$ on $l_1$ and $Q$ on $l_2$ are such that $PQ$ is perpendicular to both $l_1$ and $l_2$.

Show that

$(i)$ for all values of t the length of $PQ$ is $2\sqrt3$

$(ii)$ There is no value for of $t$ for which the planes $APB$ and $APQ$ are perpendicular to each other.

$(iii)$ The perpendicular distance from the point $A$ to the plane $BPQ$ is denoted by $p$. Show that $(27-2p^2)t^2 +(36-4p^2)t + 12 - 8p^2=0$

I was able to do part $(i)$ by first calculating the vector product of the direction vectors of both the lines, and then using the vector product as the normal and taking the point $A$, calculate the equation of plane. Then I used point B in the point to plane perpendicular distance formula to show the given result. I have one confusion in part one. What does is the significance (geometric or otherwise) of the phrase 'for all values of t'. For part $(ii)$ I have an idea about where to start i.e. I have guessed that the solution must come from here:- that when I dot the normals of $APB$ and $APQ$ and set them equal to zero, I will get an equation in $t$ that has no real roots. However this is just an idea and I was unable to move forward with it. I was no idea how to do part three. Any help would be appreciated

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  • $\begingroup$ The idea here is that your lines are varying as functions of time ($t$). So the significance is that the distance between these points P and Q is constant regardless of the time $\endgroup$ – Alan Dec 15 '14 at 17:12
  • $\begingroup$ how do we know that t here denotes time? $\endgroup$ – user140161 Dec 15 '14 at 17:14
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    $\begingroup$ It's a general context thing. You can interpret $t$ as any real value parameter, just makes it easier to visualize it as time $\endgroup$ – Alan Dec 15 '14 at 17:15
  • $\begingroup$ what exactly is a parameter? $\endgroup$ – user140161 Dec 15 '14 at 18:30

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