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Math Help - Linear Algebra understanding question

  1. #1
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    Linear Algebra understanding question

    given that \vec{v}\in \mathbb{R}^4
    and meets the requirements:
    \vec{v}\cdot \vec{e_{2}}<0 ,
     (\vec{e}_2+\vec{e}_4)\cdot \vec{v}=0
     \vec{e}_1 \cdot \vec{v}=\vec{e}_4\cdot \vec{v}
    (\vec{e}_2+\vec{e}_3)\cdot \vec{v}=0
    \left \| \vec{v} \right \| = 6

    Find the vector \vec{v} (If exists)

    Can some one please help me here?
    Thank you!
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  2. #2
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    If we write the vector as (v_1,v_2,v_3,v_4), the 5 conditions can be written as follows:

    v_2<0

    v_2+v_4=0

    v_1=v_4

    v_2+v_3=0

    v_1^2+v_2^2+v_3^2+v_4^2=36

    Letting v_4=x, the first four conditions give us that v_1=v_3=v_4=x and v_2=-x where x>0

    By the last condition, 4x^2=36 so that x=\pm 3

    We reject the negative because x must be positive

    So the vector is (3,-3,3,3)
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  3. #3
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    Thank you!
    can you please explain why?
    what the meaning of \vec{e}, Unit vector?
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  4. #4
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    Yes. For example \vec{e_1}=(1,0,0,0)
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  5. #5
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    Hooo i see...
    Thank you very much!!!
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