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Math Help - Lines and planes question

  1. #1
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    Lines and planes question

    If a plane is defined by two lines, and I have one line, like this

    r=(a,b,c) + t (d,e,f)

    Then can I just arbitrarily add other direction vectors and all of those new planes must intersect at that line?

    r=(a,b,c) + t (d,e,f) + s(5,4,7)
    r=(a,b,c) + t (d,e,f) + s(9,1,3)

    etc...
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  2. #2
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    Quote Originally Posted by theowne View Post
    If a plane is defined by two lines, and I have one line, like this r=(a,b,c) + t (d,e,f)
    Then can I just arbitrarily add other direction vectors and all of those new planes must intersect at that line?
    r=(a,b,c) + t (d,e,f) + s(5,4,7)
    r=(a,b,c) + t (d,e,f) + s(9,1,3)
    This appears to be a continuation of a previous question. I was reluctant to try to answer because the way you posted it shows some real confusion in concepts.
    First, l(t) = \left\langle {a,b,c} \right\rangle  + t\left\langle {d,e,f} \right\rangle is a line.
    Second, \pi _1 (t,s) = \left\langle {a,b,c} \right\rangle  + t\left\langle {d,e,f} \right\rangle  + s\left\langle {h,j,k} \right\rangle is a plane provided that the vectors \left\langle {d,e,f} \right\rangle \,\& \,\left\langle {h,j,k} \right\rangle are independent, (i.e. they are not parallel, that is they are not multiples of each other).
    Note that if s=0 then l(t) \subseteq \pi _1 (t,s).
    Thus, you are correct both planes \pi _1 (t,s) = \left\langle {a,b,c} \right\rangle  + t\left\langle {d,e,f} \right\rangle  + s\left\langle {5,4,7} \right\rangle \,\& \,\pi _2 (t,s) = \left\langle {a,b,c} \right\rangle  + t\left\langle {d,e,f} \right\rangle  + s\left\langle {9,1,3} \right\rangle contain the line l(t).
    But note that you must pick vectors that are independent of the direction vector.
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  3. #3
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    Well, let's say the you're given a line,
    r=(1,2,3) + t(4,5,6)

    If you have another direction vector, you can define a plane, right?

    eg

    =(1,2,3) + t(4,5,6) + s(5,4,7)

    This is a plane which passes through the above line, right?

    How about if arbitrarily replace the above s vector with another, like

    =(1,2,3) + t(4,5,6) + s(9,1,3)

    These both pass through the mentioned line, then? So both of these planes intersect at that line? Is that all there is to it?
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