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Math Help - 2(cd) intersecting lines

  1. #1
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    Joined
    Feb 2010
    Posts
    93

    2(cd) intersecting lines

    wait wait!
    I got values of μ and λ from the earlier post.
    I compared i and j components and substituted back into equation 1

    r1=[j+ak] +λ[2i+4j-3k]
    r2=[4i+3j-6k]+μ[i+5j+k]

    I need to rewrite these in parametric form

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

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

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

    (1)=(2)
    4+μ=(2+5μ)/4
    16+4μ=2+5μ
    μ=14

    sub into (1)
    λ=4+μ
    λ=4+14
    λ=18

    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=18
    μ=14

    LHS a-3λ=μ-6
    a-3(18)=14-6
    a-54=8
    a=62

    r2=4i+3j-6k+μ[i+5j+k]
    r2=4i+3j-6k+14[i+5j+k]
    r2=4i+3j-6k+14i+70j+14k
    18i+73j-8k

    coordinates of point of intersection are:
    (18, 73, -8)

    How's that?
    Last edited by stealthmaths; February 21st 2010 at 01:43 PM. Reason: made an error
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  2. #2
    Member
    Joined
    Feb 2010
    Posts
    93
    wait wait!
    I got values of μ and λ from the earlier post.
    I compared i and j components and substituted back into equation 1

    r1=[j+ak] +λ[2i+4j-3k]
    r2=[4i+3j-6k]+μ[i+5j+k]

    I need to rewrite these in parametric form

    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)

    Somebody pleeeease tell me this is correct?
    Last edited by stealthmaths; February 21st 2010 at 03:09 PM. Reason: errors
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