If ([2 p -3 r ~ q ~ s ]+[3 p +2 q ~ r ~ s ]= m [ p ~ r ~ s ]+ n [ q ~ r ~ s ]+ t [ p ~ q ~ s ] ),, then the values of m, n, t respectively are ....

Step-by-step solution

Given: [2 p - 3 r q s ] + [3 p + 2 q r s ] = m[ p r s ] + n[ q r s ] + t[ p q s ] Using linearity of scalar triple products: [ a + b c d ] = [ a c d ] + [ b c d ] . Expand the first term: [2 p - 3 r q s ] = 2[ p q s ] - 3[ r q s ] Since [ r q s ] = -[ q r s ] , this simplifies to 2[ p q s ] + 3[ q r s ] . Similarly, the second term expands to 3[ p r s ] + 2[ q r s ] . Combining both terms gives the left-hand side: LHS = 3[ p r s ] + 5[ q r s ] + 2[ p q s ] Matching coefficients with the right-hand side yields m = 3