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Math Help - Orthonormal and orthogonal vectors

  1. #1
    Member mybrohshi5's Avatar
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    Orthonormal and orthogonal vectors

    Let

     V_1 = \begin{bmatrix}-0.5\\-0.5\\0.5\\-0.5 \end{bmatrix}  V_2 = \begin{bmatrix}-0.5\\0.5\\0.5\\0.5 \end{bmatrix}  V_3 = \begin{bmatrix}0.5\\0.5\\0.5\\-0.5 \end{bmatrix}

    Find a vector in such that the vectors , , , and are orthonormal.

     V_4 = \begin{bmatrix}.\\.\\.\\.\end{bmatrix}


    I know orthogonal vectors are those that if multiplied together equal 0.

    I know that orthonormal vectors are vectors that are orthogonal and have a unit length of 1.

    So V_4 will have to be able to be multiplied by V_1 or V_2 or V_3 and equal 0 in order to be orthogonal and then it must have a unit length of 1.

    I just am not sure how to solve this type of problem

    Thank you for any help
    Last edited by mybrohshi5; April 16th 2010 at 02:42 PM.
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  2. #2
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    Let V_4 = (a,b,c,d)^t and solve the simultaneous linear equations \langle V_1, V_4 \rangle = \langle V_2, V_4 \rangle = \langle V_3, V_4 \rangle = 0. You should then have a,b,c,d in terms of one of those variables, a say. Then solve \langle V_4, V_4 \rangle = 1.
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  3. #3
    Member mybrohshi5's Avatar
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    Quote Originally Posted by Giraffro View Post
    Let V_4 = (a,b,c,d)^t and solve the simultaneous linear equations \langle V_1, V_4 \rangle = \langle V_2, V_4 \rangle = \langle V_3, V_4 \rangle = 0. You should then have a,b,c,d in terms of one of those variables, a say. Then solve \langle V_4, V_4 \rangle = 1.

    I am a little confused

    Can you get me started on how to solve the simultaneous linear equations

    \langle V_1, V_4 \rangle = \langle V_2, V_4 \rangle = \langle V_3, V_4 \rangle = 0

    I am having a brain fart when thinking about how to go about this

    This is what i have so far.

    V_1*V_4 ---> \begin{bmatrix}-0.5\\-0.5\\0.5\\-0.5\end{bmatrix} * \begin{bmatrix}a\\b\\c\\d\end{bmatrix} = 0

    -\frac{1}{2}a + -\frac{1}{2}b + \frac{1}{2}c + -\frac{1}{2}d = 0

    -1a + -1b + 1c = d

    I am not sure what to do from here now or if this is even right so far
    Last edited by mybrohshi5; April 17th 2010 at 11:15 AM.
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  4. #4
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    Quote Originally Posted by mybrohshi5 View Post
    I am a little confused

    Can you get me started on how to solve the simultaneous linear equations

    \langle V_1, V_4 \rangle = \langle V_2, V_4 \rangle = \langle V_3, V_4 \rangle = 0

    I am having a brain fart when thinking about how to go about this

    This is what i have so far.

    V_1*V_4 ---> \begin{bmatrix}-0.5\\-0.5\\0.5\\-0.5\end{bmatrix} * \begin{bmatrix}a\\b\\c\\d\end{bmatrix} = 0

    -\frac{1}{2}a + -\frac{1}{2}b + \frac{1}{2}c + -\frac{1}{2}d = 0

    -1a + -1b + 1c = d

    I am not sure what to do from here now or if this is even right so far
    You now have one variable in terms of the other 3, so carry on with  \langle V_2, V_4 \rangle = -\frac{a}{2} + \frac{b}{2} + \frac{c}{2} + \frac{d}{2} and substitute in -a-b+c for d. Then find one variable in terms of the other 2 and so on.
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