Does $\vec{a} \times \vec b=\vec a \times\vec c$ and $\vec{a} \cdot \vec b=\vec a \cdot\vec c$ imply that $\vec b = \vec c$? Does $\vec{a}  \times \vec b=\vec a \times\vec c$ and  $\vec{a} \cdot \vec b= \vec a \cdot\vec c$ imply that $\vec b = \vec c$ if $\vec a \not=0$ ?
My attempt-
$\vec{a}  \times {(\vec b- \vec c)} = 0 \implies$ they are parallel
$\vec{a} \cdot ( \vec b-\vec c) = 0 \implies $ they are perpendicular
Thus we can conclude $\vec b = \vec c$
Is my proof correct?

Yes you are right. Both the vector equations to be true vector b and c must be the same.
