prove the vector equation

Question

Given that a,b,c ,n are vectors .

Then if n.a=0 , n.b=0 ad n.c=0 for some non zero vector n .

Then how can we prove that scalar triple product of a,b,c is zero that is [abc]=0 ?

Answer 1

If the dot product of all $a,b,c$ with $n$ are all zero then $a,b,c$ lie in the same plane i.e. they are coplanar. Correct? they are all perpindicular to some non zero vector $n$... The magnitude of the scalar triple product is the volume of some parallelopiped. This will always be zero if $a,b,c$ are coplanar. In other words they are linearly dependent and you can write $\mathbf{a} = \alpha \mathbf{b} + \beta\mathbf{c}$. Voila!

Answer 2

$\renewcommand{\vec}[1]{\boldsymbol{\mathbf{#1}}}$ Without loss of generality we can assume that $\vec a,\vec b,\vec c$ are pairwise not parallel. If $\vec{n} \cdot \vec{b} = 0, \vec{n} \cdot \vec{c} = 0$, then $\vec{n}$ is orthogonal to both $\vec{b}$ and $\vec{c}$ so $\vec{n}$ is parallel to $\vec{b} \times \vec{c}$. Since $\vec{n} \cdot \vec{a} = 0$, $\vec{a}$ is orthogonal to $\vec{b} \times \vec{c}$ and the scalar triple product is $0$.