# null vector

• Sep 28th 2012, 11:36 AM
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.
• Sep 28th 2012, 11:43 AM
TheEmptySet
Re: null vector
Quote:

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!
• Sep 28th 2012, 12:27 PM
HallsofIvy
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.