Given, $p , q$ and $r$ are non-zero coplanar vectors.
$\therefore [ p + q - r \,\,\, p - q \,\,\, q - r ]$
$=( p + q - r ) \cdot[( p - q ) \times( q - r )]$
$=( p + q - r ) \cdot[ p \times q - p \times r - q \times q + q \times r ]$
$=( p + q - r ) \cdot[ p \times q - p \times r + q \times r ](\because q \times q =0)$
$= p \cdot( p \times q )- p \cdot( p \times r )+ p \cdot( q \times r )$
$+ q \cdot( p \times q )- q \cdot( p \times r )+ q \cdot( q \times r )$
$- r \cdot( p \times q )- r \cdot( p \times r )+ r \cdot( q \times r ) $
$=[ p \,\,\, q\,\,\, r ]-[ q\,\,\, p\,\,\, r ]-[ r\,\,\, p \,\,\, q ]$
$=[ p \,\,\, q \,\,\, r ]+[ p\,\,\, q \,\,\, r ]-[ p\,\,\, q \,\,\, r ] $
$= 2 [ p\,\,\, q \,\,\, r ]-[ p\,\,\, q \,\,\, r ]=[ p\,\,\, q \,\,\, r ]$