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Math Help - lines in space

  1. #1
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    lines in space

    For the life of me; this whole topic makes no sense at all !

    The parametric vector form of the line L1 is given as r1 = u1 + rv1 ( is a real number)
    where 1 u is the position vector of 1 P1 = (1,1,−3) and v1 = P1P2where P2 = (3,3,−2) .


    The parametric vector form of the line L2 is given as r2 = u2 + sv2 (s is a real number)
    where u2 is the position vector of P3 = (−2,0,2) and v2 = −j− k .


    (a) Give the parametric scalar equations of the lines L1 and L2

    (b) Find the unit vector n with negative i component which is perpendicular to both L1and L2

    (c) The shortest distance between two lines is the length of a vector that connects the two lines and is perpendicular to both lines. For L1 and L2
    this is expressed in the vector equation r2 −r1 = tn where (t is a real number) is a
    parameter. Write this equation as 3 scalar equations and hence obtain a system of three linear equations for the three parameters r, s and t .

    (d) Solve this system of equations for r, s and t and hence find the shortest distance between the two lines L2 and L2

    (e) Find the point Q on line L1 which is closest to line L2


    Thanks for any help !
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  2. #2
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    Re: lines in space

    Quote Originally Posted by bluelavae View Post

    The parametric vector form of the line L1 is given as r1 = u1 + rv1 ( is a real number)
    where 1 u is the position vector of 1 P1 = (1,1,−3) and v1 = P1P2where P2 = (3,3,−2) .
    I completely lost with your notation. Here is what I taught for over 30yrs in the US.
    Vector form: <1,2,4>+t<-2,5,3>
    parametric form: \left\{ {\begin{array}{*{20}{l}}  {x = 1 - 2t} \\   {y = 2 + 5t} \\   {z = 4 + 3t} \end{array}} \right.
    symmetric form: \dfrac{x-1}{-2}=\dfrac{y-2}{5}=\dfrac{z-4}{3}.

    That is three form of the same line through point (1,2,4) with direction <-2,5,3>.

    Can you say what your question means?
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  3. #3
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    Re: lines in space

    Ah sorry, hopefully this image shows

    lines in space-math2.png
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  4. #4
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    Re: lines in space

    Quote Originally Posted by bluelavae View Post
    Ah sorry, hopefully this image shows
    Click image for larger version. 

Name:	math2.png 
Views:	4 
Size:	97.5 KB 
ID:	23769
    Apparently the mathematics community in your part of the world use a different set of terms.
    I do not understand it.

    But if L_1:P+tD~\&~L_2:Q+sE are two non-parallel lines then the distance between them is:
    \dfrac{{\left| {\overrightarrow {PQ}  \cdot \left( {D \times E} \right)} \right|}}{{\left\| {\left( {D \times E} \right)} \right\|}}
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