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Math Help - Matrices: 2 lines intersect

  1. #1
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    Matrices: 2 lines intersect

    Okay, so here's the question:
    L1:[ 1 2 1 ]^T +s[ 2 -1 1 ] ^T
    L2: [ 3 0 1 ]^T +t[ 1 1 2 ]^T
    I need to find where they intersect and the cos of the acute angle theta between the lines.
    I was thinking of trying to put them into matrices and then trying to equate them, but I don't really know if that's allowed.
    I was thinking
    L1=[-2 1
    1 2
    -1 1]
    And
    L2= [-1 3
    -1 0
    -2 1]

    I was hoping L2-L1 would equal 0, but I have a feeling this is wrong because L1 has the parameter s and L2 has the parameter t.
    Any hints are appreciated!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by sberxa View Post
    Okay, so here's the question:
    L1:[ 1 2 1 ]^T +s[ 2 -1 1 ] ^T
    L2: [ 3 0 1 ]^T +t[ 1 1 2 ]^T
    I need to find where they intersect and the cos of the acute angle theta between the lines.
    I was thinking of trying to put them into matrices and then trying to equate them, but I don't really know if that's allowed.
    I was thinking
    L1=[-2 1
    1 2
    -1 1]
    And
    L2= [-1 3
    -1 0
    -2 1]

    I was hoping L2-L1 would equal 0, but I have a feeling this is wrong because L1 has the parameter s and L2 has the parameter t.
    Any hints are appreciated!
    you could solve it using matrices, a bit unconventional for this kind of problem, but you can use it. here is the system you want to solve

    L_1:~~~\left< 1 + 2s, 2 - s, 1 + s \right>
    L_2:~~~\left< 3 + t, t, 1 + 2t \right>

    we want the lines to be equal component-wise in order for them to intersect. thus we need to solve the system

    1 + 2s = 3 + t .................(1)
    2 - s = t ..........................(2)
    1 + s = 1 + 2t .................(3)

    so solve that for s and t. you can use matrices if you wish, but as i said. a bit conventional for problems like these...unless this is for a linear algebra class, if it is, use matrices. if it isn't, use another method
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  3. #3
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    Thanks.
    So I found the point of intersection to be (11/3,2/3,7/3) using matrices. But how do you find the cosine of theta at these two points? If there was only an x and y value, I guess I could just have made 2 triangles and found the theta of those angles at that point. But how do you do this when there is an (x,y,z) value? Is there a formula to find the distance between two points for a 3D plane? Very confused. Thanks.
    Last edited by sberxa; October 16th 2008 at 05:41 AM.
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