Further Mathematics Vector A level, How to solve it












0












$begingroup$


The line $L_1$ is parallel to the vector $i-2j-3k$ and passes through $A$, whose position vector is $3i+3j-4k$.



The line $L_2$ is parallel to the vector $-2i+j+3k$ and passes through the point $B$, whose position vector is $-3i-j+2k$.



The point $P$ on $L_1$ and the point $Q$ on $L_2$ are such that $PQ$ is perpendicular to both $L_1$ and $L_2$.



Find:




  1. The length of $PQ$

  2. The cartesian equation of the plane $PI$ contaning $PQ$ and $L_2$

  3. The perpendocular distance of $A$ from $PI$






vectors







share|cite|improve this question











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  • $begingroup$
    What is the question, exactly?
    $endgroup$
    – mlc
    Mar 22 '17 at 5:53






  • 1




    $begingroup$
    Welcome to math.SE: you may find it useful to know a few things. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Proper formatting is expected; for some basic information about writing math at this site see e.g. here, here, here and here.
    $endgroup$
    – mlc
    Mar 22 '17 at 5:54










  • $begingroup$
    Find i) the length of PQ
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:14










  • $begingroup$
    ii) The cartesian equation of the Plane PI containing PQ and L2
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:15










  • $begingroup$
    iii) The perpendicular distance of A from PI
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:15
















0












$begingroup$


The line $L_1$ is parallel to the vector $i-2j-3k$ and passes through $A$, whose position vector is $3i+3j-4k$.



The line $L_2$ is parallel to the vector $-2i+j+3k$ and passes through the point $B$, whose position vector is $-3i-j+2k$.



The point $P$ on $L_1$ and the point $Q$ on $L_2$ are such that $PQ$ is perpendicular to both $L_1$ and $L_2$.



Find:




  1. The length of $PQ$

  2. The cartesian equation of the plane $PI$ contaning $PQ$ and $L_2$

  3. The perpendocular distance of $A$ from $PI$






vectors







share|cite|improve this question











$endgroup$












  • $begingroup$
    What is the question, exactly?
    $endgroup$
    – mlc
    Mar 22 '17 at 5:53






  • 1




    $begingroup$
    Welcome to math.SE: you may find it useful to know a few things. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Proper formatting is expected; for some basic information about writing math at this site see e.g. here, here, here and here.
    $endgroup$
    – mlc
    Mar 22 '17 at 5:54










  • $begingroup$
    Find i) the length of PQ
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:14










  • $begingroup$
    ii) The cartesian equation of the Plane PI containing PQ and L2
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:15










  • $begingroup$
    iii) The perpendicular distance of A from PI
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:15














0












0








0





$begingroup$


The line $L_1$ is parallel to the vector $i-2j-3k$ and passes through $A$, whose position vector is $3i+3j-4k$.



The line $L_2$ is parallel to the vector $-2i+j+3k$ and passes through the point $B$, whose position vector is $-3i-j+2k$.



The point $P$ on $L_1$ and the point $Q$ on $L_2$ are such that $PQ$ is perpendicular to both $L_1$ and $L_2$.



Find:




  1. The length of $PQ$

  2. The cartesian equation of the plane $PI$ contaning $PQ$ and $L_2$

  3. The perpendocular distance of $A$ from $PI$






vectors







share|cite|improve this question











$endgroup$




The line $L_1$ is parallel to the vector $i-2j-3k$ and passes through $A$, whose position vector is $3i+3j-4k$.



The line $L_2$ is parallel to the vector $-2i+j+3k$ and passes through the point $B$, whose position vector is $-3i-j+2k$.



The point $P$ on $L_1$ and the point $Q$ on $L_2$ are such that $PQ$ is perpendicular to both $L_1$ and $L_2$.



Find:




  1. The length of $PQ$

  2. The cartesian equation of the plane $PI$ contaning $PQ$ and $L_2$

  3. The perpendocular distance of $A$ from $PI$






vectors




vectors






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 22 '17 at 9:08









lioness99a

3,9012727




3,9012727










asked Mar 22 '17 at 5:48









RubberRubber

11




11












  • $begingroup$
    What is the question, exactly?
    $endgroup$
    – mlc
    Mar 22 '17 at 5:53






  • 1




    $begingroup$
    Welcome to math.SE: you may find it useful to know a few things. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Proper formatting is expected; for some basic information about writing math at this site see e.g. here, here, here and here.
    $endgroup$
    – mlc
    Mar 22 '17 at 5:54










  • $begingroup$
    Find i) the length of PQ
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:14










  • $begingroup$
    ii) The cartesian equation of the Plane PI containing PQ and L2
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:15










  • $begingroup$
    iii) The perpendicular distance of A from PI
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:15


















  • $begingroup$
    What is the question, exactly?
    $endgroup$
    – mlc
    Mar 22 '17 at 5:53






  • 1




    $begingroup$
    Welcome to math.SE: you may find it useful to know a few things. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Proper formatting is expected; for some basic information about writing math at this site see e.g. here, here, here and here.
    $endgroup$
    – mlc
    Mar 22 '17 at 5:54










  • $begingroup$
    Find i) the length of PQ
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:14










  • $begingroup$
    ii) The cartesian equation of the Plane PI containing PQ and L2
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:15










  • $begingroup$
    iii) The perpendicular distance of A from PI
    $endgroup$
    – Rubber
    Mar 22 '17 at 6:15
















$begingroup$
What is the question, exactly?
$endgroup$
– mlc
Mar 22 '17 at 5:53




$begingroup$
What is the question, exactly?
$endgroup$
– mlc
Mar 22 '17 at 5:53




1




1




$begingroup$
Welcome to math.SE: you may find it useful to know a few things. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Proper formatting is expected; for some basic information about writing math at this site see e.g. here, here, here and here.
$endgroup$
– mlc
Mar 22 '17 at 5:54




$begingroup$
Welcome to math.SE: you may find it useful to know a few things. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Proper formatting is expected; for some basic information about writing math at this site see e.g. here, here, here and here.
$endgroup$
– mlc
Mar 22 '17 at 5:54












$begingroup$
Find i) the length of PQ
$endgroup$
– Rubber
Mar 22 '17 at 6:14




$begingroup$
Find i) the length of PQ
$endgroup$
– Rubber
Mar 22 '17 at 6:14












$begingroup$
ii) The cartesian equation of the Plane PI containing PQ and L2
$endgroup$
– Rubber
Mar 22 '17 at 6:15




$begingroup$
ii) The cartesian equation of the Plane PI containing PQ and L2
$endgroup$
– Rubber
Mar 22 '17 at 6:15












$begingroup$
iii) The perpendicular distance of A from PI
$endgroup$
– Rubber
Mar 22 '17 at 6:15




$begingroup$
iii) The perpendicular distance of A from PI
$endgroup$
– Rubber
Mar 22 '17 at 6:15










1 Answer
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0












$begingroup$

Write equations of the lines in parametric form: $x_1=m_1t_1+b_1$ and $x_2=m_2t_2+b_2$ where $m_1=[1 ,,-2,,-3]^T$, $m_2=[-2,,1,,3]^T$, $b_1=[3 ,,3,,-4]^T$ and $b_2=[-3,,-1,,2]^T$. $t_1$ and $t_2$ are scalars. The distance $d$ between the lines is $$d^2=(x_1-x_2) bullet (x_1-x_2) = (x_1-x_2)^T(x_1-x_2)$$ which, if it is minimized, will give you $P$ and $Q$.



begin{align}
{partial over partial t_1}d^2&=(x_1-x_2)^Tm_1+m_1^T(x_1-x_2)=2m_1^T(x_1-x_2) \
-{partial over partial t_2}d^2&=(x_1-x_2)^Tm_2+m_2^T(x_1-x_2)=2m_2^T(x_1-x_2)
end{align}



Now set these partials equal zero and solve for $t_1$ and $t_2$. If $t_1'$ and $t_2'$ are the solution, then $P=m_1t_1'+b_1$ and $Q=m_2t_2'+b_2$. I get $t_1'=4.2222$ and $t_2'=-5.7778$.



You should be able to complete the answer with a few web searches, e.g. "plane from 3 points", etc.






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    1 Answer
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    active

    oldest

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    0












    $begingroup$

    Write equations of the lines in parametric form: $x_1=m_1t_1+b_1$ and $x_2=m_2t_2+b_2$ where $m_1=[1 ,,-2,,-3]^T$, $m_2=[-2,,1,,3]^T$, $b_1=[3 ,,3,,-4]^T$ and $b_2=[-3,,-1,,2]^T$. $t_1$ and $t_2$ are scalars. The distance $d$ between the lines is $$d^2=(x_1-x_2) bullet (x_1-x_2) = (x_1-x_2)^T(x_1-x_2)$$ which, if it is minimized, will give you $P$ and $Q$.



    begin{align}
    {partial over partial t_1}d^2&=(x_1-x_2)^Tm_1+m_1^T(x_1-x_2)=2m_1^T(x_1-x_2) \
    -{partial over partial t_2}d^2&=(x_1-x_2)^Tm_2+m_2^T(x_1-x_2)=2m_2^T(x_1-x_2)
    end{align}



    Now set these partials equal zero and solve for $t_1$ and $t_2$. If $t_1'$ and $t_2'$ are the solution, then $P=m_1t_1'+b_1$ and $Q=m_2t_2'+b_2$. I get $t_1'=4.2222$ and $t_2'=-5.7778$.



    You should be able to complete the answer with a few web searches, e.g. "plane from 3 points", etc.






    share|cite|improve this answer











    $endgroup$


















      0












      $begingroup$

      Write equations of the lines in parametric form: $x_1=m_1t_1+b_1$ and $x_2=m_2t_2+b_2$ where $m_1=[1 ,,-2,,-3]^T$, $m_2=[-2,,1,,3]^T$, $b_1=[3 ,,3,,-4]^T$ and $b_2=[-3,,-1,,2]^T$. $t_1$ and $t_2$ are scalars. The distance $d$ between the lines is $$d^2=(x_1-x_2) bullet (x_1-x_2) = (x_1-x_2)^T(x_1-x_2)$$ which, if it is minimized, will give you $P$ and $Q$.



      begin{align}
      {partial over partial t_1}d^2&=(x_1-x_2)^Tm_1+m_1^T(x_1-x_2)=2m_1^T(x_1-x_2) \
      -{partial over partial t_2}d^2&=(x_1-x_2)^Tm_2+m_2^T(x_1-x_2)=2m_2^T(x_1-x_2)
      end{align}



      Now set these partials equal zero and solve for $t_1$ and $t_2$. If $t_1'$ and $t_2'$ are the solution, then $P=m_1t_1'+b_1$ and $Q=m_2t_2'+b_2$. I get $t_1'=4.2222$ and $t_2'=-5.7778$.



      You should be able to complete the answer with a few web searches, e.g. "plane from 3 points", etc.






      share|cite|improve this answer











      $endgroup$
















        0












        0








        0





        $begingroup$

        Write equations of the lines in parametric form: $x_1=m_1t_1+b_1$ and $x_2=m_2t_2+b_2$ where $m_1=[1 ,,-2,,-3]^T$, $m_2=[-2,,1,,3]^T$, $b_1=[3 ,,3,,-4]^T$ and $b_2=[-3,,-1,,2]^T$. $t_1$ and $t_2$ are scalars. The distance $d$ between the lines is $$d^2=(x_1-x_2) bullet (x_1-x_2) = (x_1-x_2)^T(x_1-x_2)$$ which, if it is minimized, will give you $P$ and $Q$.



        begin{align}
        {partial over partial t_1}d^2&=(x_1-x_2)^Tm_1+m_1^T(x_1-x_2)=2m_1^T(x_1-x_2) \
        -{partial over partial t_2}d^2&=(x_1-x_2)^Tm_2+m_2^T(x_1-x_2)=2m_2^T(x_1-x_2)
        end{align}



        Now set these partials equal zero and solve for $t_1$ and $t_2$. If $t_1'$ and $t_2'$ are the solution, then $P=m_1t_1'+b_1$ and $Q=m_2t_2'+b_2$. I get $t_1'=4.2222$ and $t_2'=-5.7778$.



        You should be able to complete the answer with a few web searches, e.g. "plane from 3 points", etc.






        share|cite|improve this answer











        $endgroup$



        Write equations of the lines in parametric form: $x_1=m_1t_1+b_1$ and $x_2=m_2t_2+b_2$ where $m_1=[1 ,,-2,,-3]^T$, $m_2=[-2,,1,,3]^T$, $b_1=[3 ,,3,,-4]^T$ and $b_2=[-3,,-1,,2]^T$. $t_1$ and $t_2$ are scalars. The distance $d$ between the lines is $$d^2=(x_1-x_2) bullet (x_1-x_2) = (x_1-x_2)^T(x_1-x_2)$$ which, if it is minimized, will give you $P$ and $Q$.



        begin{align}
        {partial over partial t_1}d^2&=(x_1-x_2)^Tm_1+m_1^T(x_1-x_2)=2m_1^T(x_1-x_2) \
        -{partial over partial t_2}d^2&=(x_1-x_2)^Tm_2+m_2^T(x_1-x_2)=2m_2^T(x_1-x_2)
        end{align}



        Now set these partials equal zero and solve for $t_1$ and $t_2$. If $t_1'$ and $t_2'$ are the solution, then $P=m_1t_1'+b_1$ and $Q=m_2t_2'+b_2$. I get $t_1'=4.2222$ and $t_2'=-5.7778$.



        You should be able to complete the answer with a few web searches, e.g. "plane from 3 points", etc.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Mar 25 '17 at 19:14

























        answered Mar 24 '17 at 21:43









        T L DavisT L Davis

        22616




        22616






























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