The resultant of two vectors P→ and Q→ is R→. If the magnitude of Q→ is doubled, the new resultant vector becomes perpendicular to P→. Then, the magnitude of R→ is equal to

tan⁡90∘=2Qsin⁡θP+2Qcos⁡θ⇒∞=2Qsin⁡θP+2Qcos⁡θ⇒ P    PQcos⁡θ=0 Now, R2    =P2+Q2+2PQcos⁡θ                    =Q2+P[P+2Qcos⁡θ]           R2    =Q2⇒R=QAlternative method: R→=P→+Q→⇒P→=R→−Q→ and S→=P→+2Q→=R→−Q→+2Q→=R→+Q→Now, S→ and P→ are perpendicular (from figure), so          S→⋅P→=0⇒(R→+Q→)⋅(R→−Q→)=0⇒ R→=Q2⇒R=Q