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Math Help - Linear Algebra Help

  1. #1
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    Linear Algebra Help

    These are linear algebra problems for an advanced engineering math class.
    1. Find the angles in the triangle with these vertices:
    [2, -1, 0] , [5, -4, 3] , and [1, -3, 2]

    2. Determine the value a so that vectors
    x = 2i + aj + k and y = 4i - 2j - 2k are perpendicular. Compute x(dotproduct)y to verify the result.

    3. Compute the inner product of the following functions on the interval [-pi, pi] with n and m distinct positive integers:
    a. <sin mx, sin nx>
    b. <cos mx, cos nx>
    c. <cos mx, sin nx>
    d. <cos nx, cos nx>
    e. <sin nx, cos nx>
    Which functions are orthogonal?

    Any help would be greatly appreciated. Thanks in advance.
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  2. #2
    Super Member Aryth's Avatar
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    1. This is simple, let's say we label the vectors:

    \vec{x} = <2, -1, 0>

    \vec{y} = <5, -4, 3>

    \vec{z} = <1, -3, 2>

    Now, the formula for the cos\theta is:

    cos\theta_1 = \frac{\vec{x}\cdotp\vec{y}}{\lVert\vec{x}\rVert*\l  Vert\vec{y}\rVert}

    cos\theta_2 = \frac{\vec{y}\cdotp\vec{z}}{\lVert\vec{y}\rVert*\l  Vert\vec{z}\rVert}

    cos\theta_3 = \frac{\vec{x}\cdotp\vec{z}}{\lVert\vec{x}\rVert*\l  Vert\vec{z}\rVert}

    All you have to do is plug and chug:

    \color{red}\vec{x}\cdotp\vec{y}  = 2*5 + -1*-4 + 0*3 = 10 + 4 = \color{blue}14

    \color{red}\vec{y}\cdotp\vec{z}  = 5*1 + -4*-3 + 3*2 = 5 + 12 + 6 = \color{blue}23

    \color{red}\vec{x}\cdotp\vec{z}  = 2*1 + -1*-3 + 0*2 = 2 + 3 = \color{blue}5

    \color{red}\lVert\vec{x}\rVert  = \sqrt{2^2 + (-1)^2 + 0^2} = \color{blue}\sqrt{5}

    \color{red}\lVert\vec{y}\rVert  = \sqrt{5^2 + (-4)^2 + 3^2} = \color{blue}\sqrt{50}

    \color{red}\lVert\vec{z}\rVert  = \sqrt{1^2 + (-3)^2 + 2^2} = \color{blue}\sqrt{14}

    You can solve from there I'm sure.
    Last edited by Aryth; January 31st 2008 at 08:15 PM.
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  3. #3
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    2. Determine the value a so that vectors
    x = 2i + aj + k and y = 4i - 2j - 2k are perpendicular. Compute x(dotproduct)y to verify the result.
    2 vectors are perpendicular if their dot product is 0.

    3. Compute the inner product of the following functions on the interval [-pi, pi] with n and m distinct positive integers:
    a. <sin mx, sin nx>
    b. <cos mx, cos nx>
    c. <cos mx, sin nx>
    d. <cos nx, cos nx>
    e. <sin nx, cos nx>
    Which functions are orthogonal?
    on the interval [a,b], the inner product of f and g is \int_a^b f(x)g(x)dx. Functions are orthogonal if their inner product is 0.
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