Question

If $\overrightarrow{a,}\overrightarrow{b,}\overrightarrow{c}$are three non-coplanar vectors, such that $\overrightarrow{d}·\overrightarrow{a}=\overrightarrow{d}·\overrightarrow{b}=\overrightarrow{d}·\overrightarrow{c}=0,$ then show that $\overrightarrow{d}$ is the null vector.

Answer 1

Given that: $\overrightarrow{d}·\overrightarrow{a}=0$
so, either $\overrightarrow{d}$ = 0 or $\overrightarrow{d}\perp \overrightarrow{a}$

similarly, $\overrightarrow{d}·\overrightarrow{b}=0$
so, $\overrightarrow{d}$ = 0 or $\overrightarrow{d}\perp \overrightarrow{b}$

Also, $\overrightarrow{d}·\overrightarrow{c}=0$
so, $\overrightarrow{d}$ = 0 or $\overrightarrow{d}\perp \overrightarrow{c}$

But $\overrightarrow{d}$ cannot be perpendicular to $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ as $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar.

so, $\overrightarrow{d}$=0. $\overrightarrow{d}$ is a null vector.