First slide

Scalars and Vectors

Question

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

Moderate

A

$\sqrt{2} : \sqrt{3}$
B

$1: \sqrt{3}$
C

$1 : \sqrt{2}$
D

$\sqrt{3} : \sqrt{2}$

Solution

$\begin{array}{l} Case 1 : (2 P)^{2} = P^{2} + Q^{2} + 2 P Q \cos θ \\ Case 2 : (P)^{2} = P^{2} + Q^{2} + 2 P Q \cos (180 - θ) \end{array}$
Adding them
$\begin{array}{l} 5 P^{2} = 2 P^{2} + 2 Q^{2} \\ or 3 P^{2} = 2 Q^{2} \\ \frac{P}{Q} = \frac{\sqrt{2}}{\sqrt{3}} \end{array}$

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