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Math Help - null vector

  1. #1
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    null vector

    if a,b,c, are three non co planar vectors such that d.a=d.b=d.c=0(where d.a represent scalar product )then show that d is null vector.
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  2. #2
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    Re: null vector

    Quote Originally Posted by ayushdadhwal View Post
    if a,b,c, are three non co planar vectors such that d.a=d.b=d.c=0(where d.a represent scalar product )then show that d is null vector.
    Without additional hypothesis the statement is false!

    Consider the orthonormal basis of \mathbb{R}^4

    v_1=\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}


    v_2=\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}


    v_3=\begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}


    v_4=\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}

    Note that

    v_i \cdot v_j =0 but none of the vectors are zero!
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  3. #3
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    Re: null vector

    If you are working in three dimensions, then a, b, and c must form a basis (otherwise they would not be non-planar) so d can be written as d= xa+ yb+ zc for numbers x, y, and z. Take the dot product of d with xa+ yb+ zc.
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