Question

A what angle do the two forces (P+q)and (p-q) actso that the resultant is √3p²+√q_²

Answer 1

if A and B are two vectors at angle theta then there resultant vector has magnitde R given by
R.R=(A.A+B.B+2AB cos theta)
here in your question A=P+Q and B=P-Q and R.R= 3P.P+Q.Q let the angle between them be theta.
thus
3P.P+Q.Q=(P+Q).(P+Q) +(P-Q).(P-Q) +2(P+Q)(P-Q) cos theta
LHS=(P.P+Q.Q+2PQ) +(P.P+Q.Q-2P.Q)+2(P.P-Q.Q) cos theta
3P.P+Q.Q=2P.P+2Q.Q+2(P.P-Q.Q) cos theta
P.P-Q.Q=2(P.P-Q.Q) cos theta
cos theta =1/2
cos theta =cos 60
theta = 60
P.P means square P and Q.Q means square of Q