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Math Help - Mechanics, solving λ throught j and i

  1. #1
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    Mechanics, solving λ throught j and i

    Given that a=i-2j and b=-3i+j find λ if a+λb is parallel to -i-3j.
    please help. I have my a-level exam in 2 days and this question is killing me, plus the mechanics book doesnt even show how solve these questions.
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  2. #2
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    Quote Originally Posted by keaton View Post
    Given that a=i-2j and b=-3i+j find λ if a+λb is parallel to -i-3j.
    First, write down a+λb, namely (i–2j) + λ(–3i+j) = (1–3λ)i + (–2+λ)j. Then use the fact that if two vectors are parallel then they are scalar multiples of each other. So we must have (1–3λ)i + (–2+λ)j = μ(–i–3j) for some value of μ. Now compare coefficients of i and j on both sides, to get 1–3λ = –μ and –2+λ = –3μ. Those are simple simultaneous equations for λ and μ, which you can solve for λ.
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    Substitute th valus of a and b in a + λb.
    Collect the terms containing i and j. Take the ratio of their coefficints with that of -i - 3j. and equate them to a constant k. From that find λ. .
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  4. #4
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    Quote Originally Posted by keaton View Post
    Given that a=i-2j and b=-3i+j find λ if a+λb is parallel to -i-3j.
    please help. I have my a-level exam in 2 days and this question is killing me, plus the mechanics book doesnt even show how solve these questions.

    a+λb = (1-3λ)i + (λ-2)j = n(-i-3j) {ie any vector parallel to -i-3j}

    Compare coeffiecients:
    i's : -n = (1-3λ) (#)
    j's : -3n = λ-2 (*)

    So from (#) n = 3λ-1, substitute into (*) to get λ
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  5. #5
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    Thanks for the help.

    i used a+λb
    then written in terms of i and j gives:
    (1-3λ)i + (λ-2)j=-i-3j

    so
    (i)=-1
    (j)=-3

    to make them the same *(i) by 3 giving: (and if a is parallel to be a=b)
    3(1-3λ)=(λ-2)
    which symplifies to get:
    3-9λ=λ-2
    so λ=1/2
    Last edited by keaton; May 16th 2011 at 02:45 AM. Reason: i thought it would put the λ in for me, not just say it with a /
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  6. #6
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    Quote Originally Posted by keaton View Post
    Thanks for the help.

    i used a+λb
    then written in terms of i and j gives:
    (1-3λ)i + (λ-2)j=-i-3j No!. Those vectors are not equal, they are parallel, which means that they are scalar multiples of each other. Look at the previous comments to see how to deal with that.

    so
    (i)=-1 Please don't write anything like this in your A-level paper!
    (j)=-3 i and j are unit vectors in the directions of the coordinate axes. They are not numbers, and it makes no sense to put them equal to 1 or 3.

    to make them the same *(i) by 3 giving: (and if a is parallel to be a=b)
    3(1-3λ)=(λ-2)
    which symplifies to get:
    3-9λ=λ-2
    so λ=1/2 However, you have got the right answer, so maybe there is something in this method. But it doesn't look good to me.
    ..
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