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Question: Resultant of two vectors \(\overset{\rightarrow}{F_{1}}\) and \(\overset{\rightarrow}{F_{2}}\) is of...

Resultant of two vectors F1→\overset{\rightarrow}{F_{1}} and F2→\overset{\rightarrow}{F_{2}} is of magnitude P. If F2→\overset{\rightarrow}{F_{2}} is reversed, then resultant is of magnitude Q. What is the value of P2 + Q2 ?

A

(a) F12+F22F_{1}^{2} + F_{2}^{2}

A

(b) F12−F22F_{1}^{2} - F_{2}^{2}

A

(c) 2(F12−F22F_{1}^{2} - F_{2}^{2})

A

(d) 2(F12+F22F_{1}^{2} + F_{2}^{2})

Explanation

Solution

(d)

Q ∣F1→+F2→∣=P|\overset{\rightarrow}{F_{1}} + \overset{\rightarrow}{F_{2}}| = P

ŽF12+F22+2F1F2cos⁡θ=P2F_{1}^{2} + F_{2}^{2} + 2F_{1}F_{2}\cos\theta = P^{2} …..(1)

& ∣F1→−F2→∣=Q|\overset{\rightarrow}{F_{1}} - \overset{\rightarrow}{F_{2}}| = Q

Ž F12+F22−2F1F2cos⁡θ=Q2F_{1}^{2} + F_{2}^{2} - 2F_{1}F_{2}\cos\theta = Q^{2} …..(2)

(1) + (2), 2(F12+F22)=P2+Q2\begin{matrix} 2(F_{1}^{2} + F_{2}^{2}) = P^{2} + Q^{2} \end{matrix}