If vector(a, b, c) are three vectors such that vector (a . b) = vector (a . c)

Question

If \(\vec{a},\,\vec{b},\,\vec{c}\) are three vectors such that \(\vec{a}.\vec{b}=\vec{a}.\vec{c}\) and \(\vec{a}\times \vec{b}=\vec{a}\times \vec{c},\, \vec{a}\neq 0\) then show that \(\vec{b}=\vec{c}.\)

Answer 1

Given, \(\vec{a}.\vec{b}=\vec{a}.\vec{c}\)

But \(\vec{a}\) cannot be both parallel and perpendicular to \((\vec{b}\,-\,\vec{c}).\) Hence, \(\vec{b}\,=\vec{c}\)