If three vectors `vec A, vec B and vec C` are such that
`vec A .vec B =vec A .vec C`, `vec A xx vec B = vec A xx vec C`, vec A!= vec 0`
then prove that ` vec B =vec C`.

Explanation

Given, ` vec A . Vec B = vec A =vec A. vec C or vec A. vec B - vec A .vec C=0`1
or ` vec A. (vec B - vec C) =0` ..(i)
But ` vec A != vec 0`, so either vec B -vec C =vec 0 or vec B=vec C`
or ` vec A` is perpendicular to ` (vec B -vec C)`
Also ` vec A xxx vec A =vec A xx vvec C or or vec A xx vec B -vec A xx vec C =vec 0`
or ` vec A xxx (vec B- vec C) =vec 0` ...(ii)
But ` vec A != vec 0` therefore eigher ` vec B- vec C=vec 0` or vec B =vec C `or vec A ` is parallel to (bec B -vec C)`
But at a time `vec A` cannot be perpendicaular to ` (vec B -vec C)` and parallel to ` (vec B - vec C) . So equations (i) and (ii) will be true at a time if ` vec B = vec C`.