Two forces P and Qacting along two intersecting lines ,having resultant R, If the force Q is reversed in direction  and changed in to (R^2-P^2)/Q , show that the resultant is still of magnititude R.

The resultant of two forces P and Q acting at angle \(\theta\) is equal to (2m+1)\( \sqrt{P^2+Q^2} \)and when they acti at angle \(90^{\circ}\)-\(\theta\),the resultant is (2m- 1)\( \sqrt{P^2+Q^2} \). prove that \(tan\theta ={m-1}/{m+1}\)

The resultant of two forces P and Q acting along the line OA and OB respectively is at right angle toOA and the resultant of forses P' abd Q' actin respectively along the same straight line is right angles toOB. Show that PP'=QQ'.

The resultant  of two forces P and Q is perpendicular to P ; the resultant of P and Q' acting at same angle is perpendicular toQ' . prove that P is the geometric mean between Q and Q'.

Two forces P and Q have a resultant R. If the forces P be increased , then the new resultant bisects the angle between R and P . Find the increase in P.

The resultant R of the force P and Q makes an angle \(2\theta\) with the line of acting of P. P is now replaced by P+R , Q remaining unchanged . Show that the resultant makes an angle \(\theta\) with  P.

If the greatest  possible resultant of two forces P and Q is n times the least , showthat the angle \(\alpha\) between P and Q when their resultant is twice the squere root of their product is given by \(tan^2{\alpha}/2\) =\({1 \over n^2-2}\).

Two forces P,Q act at a point along two straight lines making an angle \(\alpha\) with each other and R is their resultant . Two other forces P', Q' acr along the same two lines and have a resultant R'. prove that the angle between the line of acting  the resultant is 

   \(\cos^{-1} \)(PP'+QQ'+(PQ'+P'Q)cos\(\alpha\))/RR'.

Two forces P and Q act an angle \(\alpha\) and having a resultant R . if each forces is increase by R , prove that the new resultant makes with R an angle 

\(\tan^{-1}\)\((P-Q)sin\alpha\over(P+Q+R+(P+Q)cos\alpha)\)

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