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Math Help - Calc III Vector Problems--Please help!!

  1. #1
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    Calc III Vector Problems--Please help!!

    Hello,

    Can somebody help me with the following two problems?

    1. Find a value of a such that the vector <a, 1,1> makes an angle of 45 degrees with the vector <1,2,1> or show that no such a exists.

    ---I started by using theta = arccos (u*v)/(absu*absv) must equal (sqrt2)/2 and attempting to solve from there...but I'm not sure if that's the best way to proceed.

    2. Let u = 3i + j; v = 5i - 2j; and w = i - j. Find scalars, a and b such that u = av +bw.

    I have no idea how to proceed with this one.

    I would greatly appreciate any help!!!
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by riverjib View Post
    Hello,

    Can somebody help me with the following two problems?
    ...

    2. Let u = 3i + j; v = 5i - 2j; and w = i - j. Find scalars, a and b such that u = av +bw.

    I have no idea how to proceed with this one.

    I would greatly appreciate any help!!!
     <br />
u = 3i + j = a(5i - 2j) + b(i - j) = av + bw<br />

    \Longleftrightarrow (1) \,5ai + bi = 3i and (2) \, -2aj - bj = j

    from (2)  b = -2a
    substitute to (1), 5a + -2a = 3 or a = 1

    so, b = -2
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  3. #3
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    For #1, solve for a in:
    \frac{{\left\langle {a,1,1} \right\rangle  \cdot \left\langle {1,2,1} \right\rangle }}{{\sqrt 6 \sqrt {a^2  + 2} }} = \frac{{a + 3}}{{\sqrt 6 \sqrt {a^2  + 2} }} = \frac{{\sqrt 2 }}{2}
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