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Let $r_{P/Q}$ be the position vector of $\overrightarrow P$ relative to vector $\overrightarrow Q$ and $v_{P/Q}$ the velocity vector of $\overrightarrow P$ relative to $\overrightarrow Q$.

Suppose $|v_Q| > |v_P|$ and you want to set the direction of $v_P$ such that $|r_{P/Q}|$ becomes minimal at some point in time. According to the text I have, doing so requires that $v_P \cdot v_{P/Q} = 0$

enter image description here

Sorry for the horrendous image but I hope the idea is clear. $v_P$ could be any direction and the blue circle represents all possible directions of $v_p$

Anyway, my problem lies in that I do not understand why this is the necessary condition for the closest approach.

Could someone enlighten me?

If you know of a resource containing information relevant to this question, that would also be great.

Edit: I would add more detail but unfortunately there isn't much more that I know. Of course there are two angles where this works and I guess you choose the one depending on the initial positions of the two objects.

Edit: I'm really sorry but I didn't label the image properly which resulted in the post being confusing

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2 Answers 2

Some hints to get you started:

First, consider a Galilean referential in which the problem is simpler and assume some fixed $v_P$, without taking care of its magnitude.

Second, use what you know of the distance between a line and a point (projection).

Third, minimize the distance between the line and point wrt $v_P$ under the contraint that $|v_P|<|v_Q|$.

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I am sure that it is $v_{P/Q}$ which is defined as $v_P - v_Q$. Should I post the example given in the book? –  user110503 Apr 18 at 9:33
    
@user110503: The formula as you give it does not seem to be very meaningful, but the problem is not hard and following the steps I gave, you should be able to solve it and compare with your book's explanations. –  Joce Apr 18 at 11:33
    
Well, my book doesn't have an explanation; just one example of it being done. Also, since the original post was not clear, are you sure the steps you give are still pertinent? I say this because $r_{P/Q}$ is something you would use to find the closest approach (in $r_{P/Q} \cdot v_{P/Q}$), but in this case I want to work out the direction to set to find the closest distance of all possible distances. –  user110503 Apr 18 at 13:18
    
I had wringly understood. Edited –  Joce Apr 18 at 14:47
    
Thank you but I still do not understand. I am not trying to minimize the distance, I'm trying to set the direction. Your post suggests to me that you are attempting to do the former. Fix $v_{P}$ i.e. consider the motion wrt to $P$. Then use the fact that the distance is minimum when $r_{Q/P}$ and $v_{Q/P}$ are perpendicular. –  user110503 Apr 18 at 15:55

If $\vec{v}_P$ is parallel to $\vec{v}_Q$ and $|\vec{v}_Q| > |\vec{v}_P|$ then the distance will always increase. So minimizing the component of speed parallel to the motion of Q is critical. This is stated as $\vec{v}_Q \cdot \vec{v}_P = 0$.

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