Question is

let BP = CQ =b and side of square be a.

we have to prove AQ is perpendicular to DP.

in vector form AQ = AB + BQ

and DP = DA + AP

taking dot product of AQ and DP

=AB.DA + AB.AP +BQ.DA + BQ.AP

now AB & DA and BQ &AP are perpendicular to each other so there dot product will be zero.

=AB.AP + BQ.DA  ( |AB|=|DA| = a , |BQ|=|AP| = a+b)

=a(a+b) - (a+b)a   ( BQ and DA are in opposite direction so a negative sign comes)

=0

since AQ.DP = 0 i.e. AQ and DP are perpendicular to each other.