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Find point where line intersects plane

  1. Dec 6, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the point as which the line intersects the given plane.


    2. Relevant equations

    Line: x = y - 1 = 2z
    Plane: 4x - y + 3z = 8

    3. The attempt at a solution

    I understand how to use the cross product, dot product, and find intercepts. This problem is in section 13.5 #45 of the James Stewart Calculus book. I understand the idea but need some help in solving the problem. Thanks!
     
  2. jcsd
  3. Dec 6, 2008 #2
    line:

    [tex]\frac{x-x_1}{a_1}=\frac{y-y_1}{a_2}=\frac{z-z_1}{a_3}[/tex]

    plane:

    [tex]Ax+By+Cz+D=0[/tex]

    line:

    [tex]x=x_1+ta_1 ; y=y_1+ta_2 ; z=z_1 + ta_3[/tex]

    Now we substitute the coordinates of the line (x,y,z) in the plane:

    [tex]A(x_1+ta_1)+B(y_1+ta_2)+C(z_1+ta_3)+D=0[/tex]

    [tex](Aa_1+Ba_2+Ca_3)t+Ax_1+By_1+Cz_1+D=0[/tex]

    Now let [tex]a=Aa_1+Ba_2+Ca_3[/tex] and

    [tex]b=Ax_1+By_1+Cz_1+D[/tex].

    we got [tex]at+b=0[/tex]

    If a≠0, t=-b/a

    So the point where the line intersects the plane is:

    [tex]M(x_1 - \frac{b}{a}a_1 , y_1 - \frac{b}{a}a_2 , z_1 - \frac{b}{a}a_3)[/tex]

    Regards.
     
    Last edited: Dec 6, 2008
  4. Dec 6, 2008 #3

    tiny-tim

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    Science Advisor
    Homework Helper

    Welcome to PF!

    Hi jdj333! Welcome to PF! :smile:

    You don't need cross and dot products for this! :wink:

    Hint: finding an intersection is just a simultaneous equations problem …

    just substitute the line equation into the plane equation (to make it all y, say), and solve. :smile:
     
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