If $\displaystyle \vec{a}, \vec{b}, \vec{c}$ are three non-zero unequal vectors such that $\displaystyle \vec{a} \cdot \vec{b}=\vec{a} \cdot \vec{c}$, then find the angle between $\displaystyle \vec{a}$ and $\displaystyle \vec{b}-\vec{c}$.

$\displaystyle \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} \Rightarrow \vec{a} \cdot \left( \vec{b} - \vec{c} \right) = 0 \Rightarrow \vec{a} = \vec{0} \; ; \; \vec{b} = \vec{c} \text{ or } \vec{a} \perp \left( \vec{b} - \vec{c} \right)$As, $\displaystyle \vec{a} \neq \vec{0}$ ; $\displaystyle \vec{b} \neq \vec{c}$ $\displaystyle \therefore$ the angle between $\displaystyle \vec{a}$ and $\displaystyle \vec{b} - \vec{c}$ is $\displaystyle \dfrac{\pi}{2}$