Reflection of a line in a plane












-1












$begingroup$



The line $l_1$ has the equation $r=(6i+2-2k)+lambda(4i+5j-k)$ and the plane $pi_1$: $2x-y+4z=4$, the line $l_2$ is the reflection of $l_1$ in the plane $pi_1$. Find the exact vector equation of line $l_2$




So the line intersects the plane when $lambda=-2$, giving the point $(-2,-8,0)$ which will be common on $l_1$ and $l_2$. But I am unsure on how to find the direction vector for $l_2$. Any help would be appreciated.










share|cite|improve this question









$endgroup$












  • $begingroup$
    Please do not reask questions; instead edit or comment.
    $endgroup$
    – quid
    Dec 26 '18 at 14:19
















-1












$begingroup$



The line $l_1$ has the equation $r=(6i+2-2k)+lambda(4i+5j-k)$ and the plane $pi_1$: $2x-y+4z=4$, the line $l_2$ is the reflection of $l_1$ in the plane $pi_1$. Find the exact vector equation of line $l_2$




So the line intersects the plane when $lambda=-2$, giving the point $(-2,-8,0)$ which will be common on $l_1$ and $l_2$. But I am unsure on how to find the direction vector for $l_2$. Any help would be appreciated.










share|cite|improve this question









$endgroup$












  • $begingroup$
    Please do not reask questions; instead edit or comment.
    $endgroup$
    – quid
    Dec 26 '18 at 14:19














-1












-1








-1


0



$begingroup$



The line $l_1$ has the equation $r=(6i+2-2k)+lambda(4i+5j-k)$ and the plane $pi_1$: $2x-y+4z=4$, the line $l_2$ is the reflection of $l_1$ in the plane $pi_1$. Find the exact vector equation of line $l_2$




So the line intersects the plane when $lambda=-2$, giving the point $(-2,-8,0)$ which will be common on $l_1$ and $l_2$. But I am unsure on how to find the direction vector for $l_2$. Any help would be appreciated.










share|cite|improve this question









$endgroup$





The line $l_1$ has the equation $r=(6i+2-2k)+lambda(4i+5j-k)$ and the plane $pi_1$: $2x-y+4z=4$, the line $l_2$ is the reflection of $l_1$ in the plane $pi_1$. Find the exact vector equation of line $l_2$




So the line intersects the plane when $lambda=-2$, giving the point $(-2,-8,0)$ which will be common on $l_1$ and $l_2$. But I am unsure on how to find the direction vector for $l_2$. Any help would be appreciated.







geometry vector-spaces vectors






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 20 '18 at 10:17









H.LinkhornH.Linkhorn

485213




485213












  • $begingroup$
    Please do not reask questions; instead edit or comment.
    $endgroup$
    – quid
    Dec 26 '18 at 14:19


















  • $begingroup$
    Please do not reask questions; instead edit or comment.
    $endgroup$
    – quid
    Dec 26 '18 at 14:19
















$begingroup$
Please do not reask questions; instead edit or comment.
$endgroup$
– quid
Dec 26 '18 at 14:19




$begingroup$
Please do not reask questions; instead edit or comment.
$endgroup$
– quid
Dec 26 '18 at 14:19










3 Answers
3






active

oldest

votes


















1












$begingroup$

Hint...Construct the line through $(6,2,-2)$ with direction the same as the normal to the plane, and find where this line meets the plane. This point will be the midpoint of the line joining $(6,2,-2)$ and its reflection in the plane. Once you have this reflection point you can form the line of reflection because you now have two points.



the answer I get is $r=-2i-8j+t(88i+103j-13k)$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    doing this I got an answer of $(-2i-8j+0k)+mu(92/21i + 206/21j +26/21k)$ which is apparently the wrong answer?
    $endgroup$
    – H.Linkhorn
    Dec 20 '18 at 12:03












  • $begingroup$
    Looks like you need to check your calculations
    $endgroup$
    – David Quinn
    Dec 20 '18 at 22:49










  • $begingroup$
    Would it be possible for someone to solve the problem and give me a work solution for me to compare against.
    $endgroup$
    – H.Linkhorn
    Dec 21 '18 at 16:28










  • $begingroup$
    I have added an answer
    $endgroup$
    – David Quinn
    Dec 21 '18 at 23:10










  • $begingroup$
    Would you be able to include a method for me to see where ive gone wrong?
    $endgroup$
    – H.Linkhorn
    Dec 22 '18 at 14:13



















0












$begingroup$

HINT



In the first place, I advise you to obtain the normal component of $(4,5,-1)$ related to the place $pi_{1}$. After so, all you have to do is to substract twice it from the original direction of $l_{1}$ in order to obtain the direction of $l_{2}$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    what do you mean by "normal component of $(4,5,−1)$ related to the place $π_1$"
    $endgroup$
    – H.Linkhorn
    Dec 24 '18 at 17:54










  • $begingroup$
    I mean the projection of $(4,5,-1)$ in the direction of $(2,-1,4)$. Precisely speaking, I am talking about the vector: $$langle(4,5,-1),(2,-1,4)rangle (2,-1,4)/lVert(2,-1,4)rVert^{2}$$
    $endgroup$
    – APC89
    Dec 24 '18 at 17:58





















0












$begingroup$

Given a plane $Pi_1$ and a line $L$



$$
Pi_1to (p-p_1)cdot vec n_1 = 0\
Lto p = p_0 +lambda vec n_2
$$



first we determine the intersection point



$$
p^* = Pi_1cap L
$$



by making



$$
(p_0-p_1+lambdavec n_2)cdot vec n_1 = 0Rightarrow lambda = -frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}
$$



then



$$
p^* = p_0-frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}vec n_2
$$



After that the reflection for $vec n_2$ regarding $Pi_1$ is obtained as follows:



$$
vec n_2 = alpha vec n_1+vec mRightarrow vec n_1cdotvec n_2 = alpha||vec n_1||^2
$$



then



$$
alpha = frac{vec n_1cdotvec n_2}{||vec n_1||^2}Rightarrow vec m = vec n_2- frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
$$



and then the reflected line is



$$
L_Rto p = p^* +lambda vec n_R
$$



with



$$
vec n_R = vec m - alpha vec n_1 = vec n_2- 2frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
$$






share|cite|improve this answer









$endgroup$














    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3047370%2freflection-of-a-line-in-a-plane%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    Hint...Construct the line through $(6,2,-2)$ with direction the same as the normal to the plane, and find where this line meets the plane. This point will be the midpoint of the line joining $(6,2,-2)$ and its reflection in the plane. Once you have this reflection point you can form the line of reflection because you now have two points.



    the answer I get is $r=-2i-8j+t(88i+103j-13k)$






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      doing this I got an answer of $(-2i-8j+0k)+mu(92/21i + 206/21j +26/21k)$ which is apparently the wrong answer?
      $endgroup$
      – H.Linkhorn
      Dec 20 '18 at 12:03












    • $begingroup$
      Looks like you need to check your calculations
      $endgroup$
      – David Quinn
      Dec 20 '18 at 22:49










    • $begingroup$
      Would it be possible for someone to solve the problem and give me a work solution for me to compare against.
      $endgroup$
      – H.Linkhorn
      Dec 21 '18 at 16:28










    • $begingroup$
      I have added an answer
      $endgroup$
      – David Quinn
      Dec 21 '18 at 23:10










    • $begingroup$
      Would you be able to include a method for me to see where ive gone wrong?
      $endgroup$
      – H.Linkhorn
      Dec 22 '18 at 14:13
















    1












    $begingroup$

    Hint...Construct the line through $(6,2,-2)$ with direction the same as the normal to the plane, and find where this line meets the plane. This point will be the midpoint of the line joining $(6,2,-2)$ and its reflection in the plane. Once you have this reflection point you can form the line of reflection because you now have two points.



    the answer I get is $r=-2i-8j+t(88i+103j-13k)$






    share|cite|improve this answer











    $endgroup$













    • $begingroup$
      doing this I got an answer of $(-2i-8j+0k)+mu(92/21i + 206/21j +26/21k)$ which is apparently the wrong answer?
      $endgroup$
      – H.Linkhorn
      Dec 20 '18 at 12:03












    • $begingroup$
      Looks like you need to check your calculations
      $endgroup$
      – David Quinn
      Dec 20 '18 at 22:49










    • $begingroup$
      Would it be possible for someone to solve the problem and give me a work solution for me to compare against.
      $endgroup$
      – H.Linkhorn
      Dec 21 '18 at 16:28










    • $begingroup$
      I have added an answer
      $endgroup$
      – David Quinn
      Dec 21 '18 at 23:10










    • $begingroup$
      Would you be able to include a method for me to see where ive gone wrong?
      $endgroup$
      – H.Linkhorn
      Dec 22 '18 at 14:13














    1












    1








    1





    $begingroup$

    Hint...Construct the line through $(6,2,-2)$ with direction the same as the normal to the plane, and find where this line meets the plane. This point will be the midpoint of the line joining $(6,2,-2)$ and its reflection in the plane. Once you have this reflection point you can form the line of reflection because you now have two points.



    the answer I get is $r=-2i-8j+t(88i+103j-13k)$






    share|cite|improve this answer











    $endgroup$



    Hint...Construct the line through $(6,2,-2)$ with direction the same as the normal to the plane, and find where this line meets the plane. This point will be the midpoint of the line joining $(6,2,-2)$ and its reflection in the plane. Once you have this reflection point you can form the line of reflection because you now have two points.



    the answer I get is $r=-2i-8j+t(88i+103j-13k)$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Dec 21 '18 at 23:09

























    answered Dec 20 '18 at 10:30









    David QuinnDavid Quinn

    24.1k21141




    24.1k21141












    • $begingroup$
      doing this I got an answer of $(-2i-8j+0k)+mu(92/21i + 206/21j +26/21k)$ which is apparently the wrong answer?
      $endgroup$
      – H.Linkhorn
      Dec 20 '18 at 12:03












    • $begingroup$
      Looks like you need to check your calculations
      $endgroup$
      – David Quinn
      Dec 20 '18 at 22:49










    • $begingroup$
      Would it be possible for someone to solve the problem and give me a work solution for me to compare against.
      $endgroup$
      – H.Linkhorn
      Dec 21 '18 at 16:28










    • $begingroup$
      I have added an answer
      $endgroup$
      – David Quinn
      Dec 21 '18 at 23:10










    • $begingroup$
      Would you be able to include a method for me to see where ive gone wrong?
      $endgroup$
      – H.Linkhorn
      Dec 22 '18 at 14:13


















    • $begingroup$
      doing this I got an answer of $(-2i-8j+0k)+mu(92/21i + 206/21j +26/21k)$ which is apparently the wrong answer?
      $endgroup$
      – H.Linkhorn
      Dec 20 '18 at 12:03












    • $begingroup$
      Looks like you need to check your calculations
      $endgroup$
      – David Quinn
      Dec 20 '18 at 22:49










    • $begingroup$
      Would it be possible for someone to solve the problem and give me a work solution for me to compare against.
      $endgroup$
      – H.Linkhorn
      Dec 21 '18 at 16:28










    • $begingroup$
      I have added an answer
      $endgroup$
      – David Quinn
      Dec 21 '18 at 23:10










    • $begingroup$
      Would you be able to include a method for me to see where ive gone wrong?
      $endgroup$
      – H.Linkhorn
      Dec 22 '18 at 14:13
















    $begingroup$
    doing this I got an answer of $(-2i-8j+0k)+mu(92/21i + 206/21j +26/21k)$ which is apparently the wrong answer?
    $endgroup$
    – H.Linkhorn
    Dec 20 '18 at 12:03






    $begingroup$
    doing this I got an answer of $(-2i-8j+0k)+mu(92/21i + 206/21j +26/21k)$ which is apparently the wrong answer?
    $endgroup$
    – H.Linkhorn
    Dec 20 '18 at 12:03














    $begingroup$
    Looks like you need to check your calculations
    $endgroup$
    – David Quinn
    Dec 20 '18 at 22:49




    $begingroup$
    Looks like you need to check your calculations
    $endgroup$
    – David Quinn
    Dec 20 '18 at 22:49












    $begingroup$
    Would it be possible for someone to solve the problem and give me a work solution for me to compare against.
    $endgroup$
    – H.Linkhorn
    Dec 21 '18 at 16:28




    $begingroup$
    Would it be possible for someone to solve the problem and give me a work solution for me to compare against.
    $endgroup$
    – H.Linkhorn
    Dec 21 '18 at 16:28












    $begingroup$
    I have added an answer
    $endgroup$
    – David Quinn
    Dec 21 '18 at 23:10




    $begingroup$
    I have added an answer
    $endgroup$
    – David Quinn
    Dec 21 '18 at 23:10












    $begingroup$
    Would you be able to include a method for me to see where ive gone wrong?
    $endgroup$
    – H.Linkhorn
    Dec 22 '18 at 14:13




    $begingroup$
    Would you be able to include a method for me to see where ive gone wrong?
    $endgroup$
    – H.Linkhorn
    Dec 22 '18 at 14:13











    0












    $begingroup$

    HINT



    In the first place, I advise you to obtain the normal component of $(4,5,-1)$ related to the place $pi_{1}$. After so, all you have to do is to substract twice it from the original direction of $l_{1}$ in order to obtain the direction of $l_{2}$.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      what do you mean by "normal component of $(4,5,−1)$ related to the place $π_1$"
      $endgroup$
      – H.Linkhorn
      Dec 24 '18 at 17:54










    • $begingroup$
      I mean the projection of $(4,5,-1)$ in the direction of $(2,-1,4)$. Precisely speaking, I am talking about the vector: $$langle(4,5,-1),(2,-1,4)rangle (2,-1,4)/lVert(2,-1,4)rVert^{2}$$
      $endgroup$
      – APC89
      Dec 24 '18 at 17:58


















    0












    $begingroup$

    HINT



    In the first place, I advise you to obtain the normal component of $(4,5,-1)$ related to the place $pi_{1}$. After so, all you have to do is to substract twice it from the original direction of $l_{1}$ in order to obtain the direction of $l_{2}$.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      what do you mean by "normal component of $(4,5,−1)$ related to the place $π_1$"
      $endgroup$
      – H.Linkhorn
      Dec 24 '18 at 17:54










    • $begingroup$
      I mean the projection of $(4,5,-1)$ in the direction of $(2,-1,4)$. Precisely speaking, I am talking about the vector: $$langle(4,5,-1),(2,-1,4)rangle (2,-1,4)/lVert(2,-1,4)rVert^{2}$$
      $endgroup$
      – APC89
      Dec 24 '18 at 17:58
















    0












    0








    0





    $begingroup$

    HINT



    In the first place, I advise you to obtain the normal component of $(4,5,-1)$ related to the place $pi_{1}$. After so, all you have to do is to substract twice it from the original direction of $l_{1}$ in order to obtain the direction of $l_{2}$.






    share|cite|improve this answer









    $endgroup$



    HINT



    In the first place, I advise you to obtain the normal component of $(4,5,-1)$ related to the place $pi_{1}$. After so, all you have to do is to substract twice it from the original direction of $l_{1}$ in order to obtain the direction of $l_{2}$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Dec 24 '18 at 17:33









    APC89APC89

    2,371720




    2,371720












    • $begingroup$
      what do you mean by "normal component of $(4,5,−1)$ related to the place $π_1$"
      $endgroup$
      – H.Linkhorn
      Dec 24 '18 at 17:54










    • $begingroup$
      I mean the projection of $(4,5,-1)$ in the direction of $(2,-1,4)$. Precisely speaking, I am talking about the vector: $$langle(4,5,-1),(2,-1,4)rangle (2,-1,4)/lVert(2,-1,4)rVert^{2}$$
      $endgroup$
      – APC89
      Dec 24 '18 at 17:58




















    • $begingroup$
      what do you mean by "normal component of $(4,5,−1)$ related to the place $π_1$"
      $endgroup$
      – H.Linkhorn
      Dec 24 '18 at 17:54










    • $begingroup$
      I mean the projection of $(4,5,-1)$ in the direction of $(2,-1,4)$. Precisely speaking, I am talking about the vector: $$langle(4,5,-1),(2,-1,4)rangle (2,-1,4)/lVert(2,-1,4)rVert^{2}$$
      $endgroup$
      – APC89
      Dec 24 '18 at 17:58


















    $begingroup$
    what do you mean by "normal component of $(4,5,−1)$ related to the place $π_1$"
    $endgroup$
    – H.Linkhorn
    Dec 24 '18 at 17:54




    $begingroup$
    what do you mean by "normal component of $(4,5,−1)$ related to the place $π_1$"
    $endgroup$
    – H.Linkhorn
    Dec 24 '18 at 17:54












    $begingroup$
    I mean the projection of $(4,5,-1)$ in the direction of $(2,-1,4)$. Precisely speaking, I am talking about the vector: $$langle(4,5,-1),(2,-1,4)rangle (2,-1,4)/lVert(2,-1,4)rVert^{2}$$
    $endgroup$
    – APC89
    Dec 24 '18 at 17:58






    $begingroup$
    I mean the projection of $(4,5,-1)$ in the direction of $(2,-1,4)$. Precisely speaking, I am talking about the vector: $$langle(4,5,-1),(2,-1,4)rangle (2,-1,4)/lVert(2,-1,4)rVert^{2}$$
    $endgroup$
    – APC89
    Dec 24 '18 at 17:58













    0












    $begingroup$

    Given a plane $Pi_1$ and a line $L$



    $$
    Pi_1to (p-p_1)cdot vec n_1 = 0\
    Lto p = p_0 +lambda vec n_2
    $$



    first we determine the intersection point



    $$
    p^* = Pi_1cap L
    $$



    by making



    $$
    (p_0-p_1+lambdavec n_2)cdot vec n_1 = 0Rightarrow lambda = -frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}
    $$



    then



    $$
    p^* = p_0-frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}vec n_2
    $$



    After that the reflection for $vec n_2$ regarding $Pi_1$ is obtained as follows:



    $$
    vec n_2 = alpha vec n_1+vec mRightarrow vec n_1cdotvec n_2 = alpha||vec n_1||^2
    $$



    then



    $$
    alpha = frac{vec n_1cdotvec n_2}{||vec n_1||^2}Rightarrow vec m = vec n_2- frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
    $$



    and then the reflected line is



    $$
    L_Rto p = p^* +lambda vec n_R
    $$



    with



    $$
    vec n_R = vec m - alpha vec n_1 = vec n_2- 2frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
    $$






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Given a plane $Pi_1$ and a line $L$



      $$
      Pi_1to (p-p_1)cdot vec n_1 = 0\
      Lto p = p_0 +lambda vec n_2
      $$



      first we determine the intersection point



      $$
      p^* = Pi_1cap L
      $$



      by making



      $$
      (p_0-p_1+lambdavec n_2)cdot vec n_1 = 0Rightarrow lambda = -frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}
      $$



      then



      $$
      p^* = p_0-frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}vec n_2
      $$



      After that the reflection for $vec n_2$ regarding $Pi_1$ is obtained as follows:



      $$
      vec n_2 = alpha vec n_1+vec mRightarrow vec n_1cdotvec n_2 = alpha||vec n_1||^2
      $$



      then



      $$
      alpha = frac{vec n_1cdotvec n_2}{||vec n_1||^2}Rightarrow vec m = vec n_2- frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
      $$



      and then the reflected line is



      $$
      L_Rto p = p^* +lambda vec n_R
      $$



      with



      $$
      vec n_R = vec m - alpha vec n_1 = vec n_2- 2frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
      $$






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Given a plane $Pi_1$ and a line $L$



        $$
        Pi_1to (p-p_1)cdot vec n_1 = 0\
        Lto p = p_0 +lambda vec n_2
        $$



        first we determine the intersection point



        $$
        p^* = Pi_1cap L
        $$



        by making



        $$
        (p_0-p_1+lambdavec n_2)cdot vec n_1 = 0Rightarrow lambda = -frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}
        $$



        then



        $$
        p^* = p_0-frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}vec n_2
        $$



        After that the reflection for $vec n_2$ regarding $Pi_1$ is obtained as follows:



        $$
        vec n_2 = alpha vec n_1+vec mRightarrow vec n_1cdotvec n_2 = alpha||vec n_1||^2
        $$



        then



        $$
        alpha = frac{vec n_1cdotvec n_2}{||vec n_1||^2}Rightarrow vec m = vec n_2- frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
        $$



        and then the reflected line is



        $$
        L_Rto p = p^* +lambda vec n_R
        $$



        with



        $$
        vec n_R = vec m - alpha vec n_1 = vec n_2- 2frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
        $$






        share|cite|improve this answer









        $endgroup$



        Given a plane $Pi_1$ and a line $L$



        $$
        Pi_1to (p-p_1)cdot vec n_1 = 0\
        Lto p = p_0 +lambda vec n_2
        $$



        first we determine the intersection point



        $$
        p^* = Pi_1cap L
        $$



        by making



        $$
        (p_0-p_1+lambdavec n_2)cdot vec n_1 = 0Rightarrow lambda = -frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}
        $$



        then



        $$
        p^* = p_0-frac{(p_0-p_1)cdotvec n_1}{vec n_1cdotvec n_2}vec n_2
        $$



        After that the reflection for $vec n_2$ regarding $Pi_1$ is obtained as follows:



        $$
        vec n_2 = alpha vec n_1+vec mRightarrow vec n_1cdotvec n_2 = alpha||vec n_1||^2
        $$



        then



        $$
        alpha = frac{vec n_1cdotvec n_2}{||vec n_1||^2}Rightarrow vec m = vec n_2- frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
        $$



        and then the reflected line is



        $$
        L_Rto p = p^* +lambda vec n_R
        $$



        with



        $$
        vec n_R = vec m - alpha vec n_1 = vec n_2- 2frac{vec n_1cdotvec n_2}{||vec n_1||^2}vec n_1
        $$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 25 '18 at 1:26









        CesareoCesareo

        9,7763517




        9,7763517






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3047370%2freflection-of-a-line-in-a-plane%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            Plaza Victoria

            Puebla de Zaragoza

            Musa