Suppose that a quadrilateral has all sides of equal length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular. Solution Let [veca and vecb ] represented by sides of quadrilateral of equal length. i.e. [|veca|=|vecb|=>|a|^2=|b|^2] Also let define [veca.veca=|a|^2] and [vecb.vecb=|b|^2] The diagonals of the quadrilateral will be represented by [veca+vecb ] [and vecb-veca.] Two vectors are perpendicular if their dot product is zero. Thus [(veca+vecb).(vecb-veca)=veca.vecb+vecb.vecb-veca.veca-vecb.veca] [since ] [veca.vecb=vecb.veca] [therefore] [(veca+vecb).(vecb-veca)=veca.vecb+|b|^2-|a|^2-veca.vecb] [=0] This implies diagonals are perpendicular..