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Homework Help: Intersecting vector lines

  1. Feb 21, 2010 #1
    1.Consider the two straight lines:
    2λi+(1+4λ)j+(3-3λ)k
    (4+μ)i+(3+5μ)j+(-6+μ)k

    a) Show lines are skew when a=3.
    b) Find value of a for which lines intersect and state coordinates of point of intersection


    2.
    2λi+(1+4λ)j+(3-3λ)k=(4+μ)i+(3+5μ)j+(-6+μ)k

    comparing i components:
    (1)
    2λ=4+μ
    λ=(4+μ)/2

    comparing j components:
    (2)
    1+4λ=3+5μ
    λ=(2+5μ)/4

    (1)=(2)
    (4+μ)/2=(2+5μ)/4
    16+4μ=4+10μ
    μ=2

    sub into (1)
    λ=(4+μ)/2
    λ=(4+2)/2
    λ=3

    Different values so the lines are skew

    d) Find a value of a for which the lines intersect and state the coordinates of the point of intersection.
    r1=2λi+(4λ+1)j+(a-3λ)k

    r2=(μ+4)i+(5μ+3)j+(μ-6)k

    I need to find the value of a for which the lines intersect and state the coordinates of point of intersection.

    λ=3
    μ=2

    LHS a-3λ=μ-6
    a-3(3)=2-6
    a-9=-4
    a=5

    r2=4i+3j-6k+μ[i+5j+k]
    r2=4i+3j-6k+2[i+5j+k]
    r2=4i+3j-6k+2i+10j+2k
    6i+13j-4k

    coordinates of point of intersection are:
    (6, 13, -4)

    could somebody tell me if i'm on the right track here please?
     
  2. jcsd
  3. Feb 22, 2010 #2
    Hi lemon


    It should be : 2λi+(1+4λ)j+(a-3λ)k

    What do you mean by "different values" ? You only find one value of λ and μ so how can you say "different values" ? To what did you compare or check it?

    The rest of your work is nice. :smile:
     
  4. Feb 22, 2010 #3
    ahh yes. I should say:
    With the values of λ and μ the coefficients of K become:
    LHS=3-3(3)=-6
    RHS=-6+2=-4
    LHS≠RHS, and so the lines are skew.


    I'm not sure how I would compare of check this?
     
    Last edited: Feb 22, 2010
  5. Feb 22, 2010 #4
    If I don't misinterpret your post, the value of λ and μ should not be the same for the line to be skew. Check it using the third equation, comparing the k components. You'll end up LHS is not the same as RHS.
     
  6. Feb 22, 2010 #5
    yes. Thank you songoku. The values are different and so the lines are skew.
     
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