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 $\displaystyle \mathbb{R}^4$
$\displaystyle v_1=\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$
$\displaystyle v_2=\begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}$
$\displaystyle v_3=\begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}$
$\displaystyle v_4=\begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}$
Note that
$\displaystyle v_i \cdot v_j =0$ but none of the vectors are zero!