Answer :

Given:- , P and Q are trisection of AB

i.e. AP = PQ = QB or 1:1:1 division of line AB


To Prove:-



Proof:- Let be position vector of O, A and B respectively


Now, Find position vector of P, we use section formulae of internal division: Theorem given below


Let A and B be two points with position vectors


respectively, and c be a point dividing AB internally in the ration m:n. Then the position vector of c is given by


By above theorem, here P point divides AB in 1:2, so we get




Similarly, Position vector of Q is calculated


By above theorem, here Q point divides AB in 2:1, so we get




Length OA and OB in vector form






Now length/distance OP in vector form









length/distance OQ in vector form









Taking LHS


OP2 + OQ2


=


=


as we know in case of dot product




Angle between OA and OB is 90°,




Therefore, OP2 + OQ2


=


=


=


=


As from figure OA2 + OB2 = AB2


=


= RHS


Hence, Proved.


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