Statement-1: The magnitude of the resultant of two vectors $P$ and $Q$ acting at an angle $θ$ is maximum when $θ = 0^{\circ}$ .
Statement-2: The magnitude of the resultant of two vectors $P$ and $Q$ is given by $P^{2} + Q^{2} + 2 PQ cos θ$ .

Question

Statement-1: The magnitude of the resultant of two vectors $P$ and $Q$ acting at an angle $θ$ is maximum when $θ = 0^{\circ}$ .

Statement-2: The magnitude of the resultant of two vectors $P$ and $Q$ is given by $P^{2} + Q^{2} + 2 PQ cos θ$ .

TestHub · Accepted Answer

Statement-2: The magnitude of the resultant vector $R = P + Q$ is given by the law of cosines: $R = P^{2} + Q^{2} + 2 PQ cos θ$ , where $P = ∣ P ∣$ and $Q = ∣ Q ∣$ . Thus, Statement-2 is true.

Statement-1: From the formula in Statement-2, $R = P^{2} + Q^{2} + 2 PQ cos θ$ . For $R$ to be maximum, the term $2 PQ cos θ$ must be maximum. Since $P$ and $Q$ are magnitudes (positive), this occurs when $cos θ$ is maximum. The maximum value of $cos θ$ is $1$ , which occurs when $θ = 0^{\circ}$ . At $θ = 0^{\circ}$ , $R_{ma x} = P^{2} + Q^{2} + 2 PQ = (P + Q)^{2} = P + Q$ . Thus, Statement-1 is true.

Statement-2 provides the formula for the resultant magnitude, which is directly used to determine the condition for maximum resultant in Statement-1. Therefore, Statement-2 is a correct explanation for Statement-1.

Physics - Vectors Question with Solution | TestHub

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