Question

When two vectors of magnitudes P and Q are inclined at an angle $θ$ , the magnitude of their resultant 2P. When the inclination is changed to 180- $θ$ , the magnitude of the resultant is halved. Find the ratio of P to Q.

Moderate

Solution

$R_{1}^{2}$ = ${(2 P)}^{2} = P^{2} + Q^{2} + 2 PQcosθ$
${(P)}^{2} = P^{2} + Q^{2} - 2 PQcosθ$
Adding them $5 P^{2} = 2 P^{2} + 2 Q^{2}$
or $3 P^{2} = 2 Q^{2} \Rightarrow \frac{P}{Q} = \frac{\sqrt{2}}{\sqrt{3}}$

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