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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- Sep 28th 2012, 10:36 AMayushdadhwalnull 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.

- Sep 28th 2012, 10:43 AMTheEmptySetRe: 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! - Sep 28th 2012, 11:27 AMHallsofIvyRe: 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.