Find closest possible points between 2 lines? (vectors)? Help!

[Mathlaber]

Find points P,Q which are closest possible with P lying on the line

x=8+1t
y=8+1t
z=7−3t

and Q lying on the line
x=231−6t
y=−10−17t
z=71−13t

 

[Ishuda]

Find points P,Q which are closest possible with P lying on the line

x=8+1t
y=8+1t
z=7−3t

and Q lying on the line
x=231−6t
y=−10−17t
z=71−13t

What are your thoughts? What have you done so far? Please show us your work even if you feel that it is wrong so we may try to help you. You might also read
http://www.freemathhelp.com/forum/threads/78006-Read-Before-Posting

As a hint, if the distance between the two points P & Q are a minimum so is the squared distance.

 

[Otis]

Ishuda may have read the exercise as asking for the shortest distance, instead of for the coordinates of the endpoints.

I'm thinking that we need separate parameters for each line, as there's no guarantee that the two closest points share the same parameter value.

P = <8, 8, 7> + t * <1, 1, -3>

Q = <231, -10, 71> + s * <-6, -17, -13>

Calculate PQ, and use the fact that PQ is orthogonal to each line. That is,

PQ * <1, 1, -3> = 0

PQ * <-6, -17, -13> = 0

You'll get a system of two equations in s and t, to solve.

Calculate the coordinates, by substituting your t in P and your s in Q.

I worked it. Wanna compare answers? :cool:

 

[Ishuda]

Ishuda may have read the exercise as asking for the shortest distance, instead of for the coordinates of the endpoints.

I'm thinking that we need separate parameters for each line, as there's no guarantee that the two closest points share the same parameter value.

P = <8, 8, 7> + t * <1, 1, -3>

Q = <231, -10, 71> + s * <-6, -17, -13>

Calculate PQ, and use the fact that PQ is orthogonal to each line. That is,

PQ * <1, 1, -3> = 0

PQ * <-6, -17, -13> = 0

You'll get a system of two equations in s and t, to solve.

Calculate the coordinates, by substituting your t in P and your s in Q.

I worked it. Wanna compare answers? :cool:

I didn't work out the answer but, yes, that is what I was thinking.

 

[pka]

Find points P,Q which are closest possible with P lying on the line
x=8+1t , y=8+1t , z=7−3t

and Q lying on the line
x=231−6t , y=−10−17t , z=71−13t

I'm thinking that we need separate parameters for each line, as there's no guarantee that the two closest points share the same parameter value.
\(\displaystyle \ell_1=\)\(\displaystyle <8, 8, 7> + t <1, 1, -3>\~&~\ell_2= <231, -10, 71> + s<-6, -17, -13>\).
and use the fact that PQ is orthogonal to each line. That is FALSE!
You'll get a system of two equations in s and t, to solve.
Calculate the coordinates, by substituting your t in P and your s in Q.

\(\displaystyle \overrightarrow {PQ} = \left\langle {223, - 18,64} \right\rangle \). See this page.

Let \(\displaystyle \overrightarrow {E} =<1,1,-3>~\&~\overrightarrow {F} =<-6,-17,-13> \)

To see that \(\displaystyle \overrightarrow {PQ}\) is not orthogonal to \(\displaystyle \ell_1\) use this page to see that \(\displaystyle \overrightarrow {PQ}\cdot E\not= 0\)

The irony of this question is that it almost trivial to find the distance between \(\displaystyle \ell_1~\&~\ell_2\).
It is \(\displaystyle \dfrac{|\overrightarrow {PQ}\cdot( E\times F)|}{\| E\times F\|}=3\sqrt{5178}\)

It is very difficult to find the coordinates of the endpoints of the unique perpendicular to both \(\displaystyle \ell_1~\&~\ell_2\).

 

[Otis]

The irony of this question is that it almost trivial to find the distance

It appears as though you have joined Ishuda in misreading the question; it does not ask for any distance.

It is very difficult to find the coordinates of the endpoints

Thank you, pka, for pointing out my goof. I knew that something extra was needed, to find coordinates, but apparently I incorrectly assumed orthogonality.

Interestingly, my incorrect method led to two points separated by a distance that matches the distance that you posted. This pair may not be unique, but it works to answer the OP.

Cheers :cool:

 

[Ishuda]

It appears as though you have joined Ishuda in misreading the question; it does not ask for any distance. ...

The problem is "Find points P,Q which are closest possible ...". One way to work the solution is to write the general formula for the distance between any two points, P on the one given line and Q on the other given line. That formula will be a function of two variables as Otis pointed out. If you minimize that function wrt s and t (using the variable names Otis used) you will obtain the s and t for "points P,Q which are closest possible."

Since the distance function contains a square root which may introduce complications [actually it does only very slightly], one might want to remove those complications by minimizing the squared distance since that will also obtain the s and t for "points P,Q which are closest possible."

 

[Otis]

Find points P,Q which are closest possible

points P,Q which are closest possible

P,Q which are closest possible

I am obviously a poor communicator.

Ishuda, you report a distance, and I will report the locations of two points, and then we'll wait to see who gets credit.

 

[Ishuda]

I am obviously a poor communicator.

Ishuda, you report a distance, and I will report the locations of two points, and then we'll wait to see who gets credit.

What do you not understand about finding the s and t which determine the points P & Q which are closest.