Question

The sum of two forces \( \overrightarrow P \) and \( \overrightarrow Q \) is \( \overrightarrow R \) such that \( \left| {\overrightarrow R } \right| = \left| {\overrightarrow P } \right| \) . The angle \( \theta \) (in degrees) that the resultant of 2\( {\overrightarrow P } \) and \( {\overrightarrow Q } \) will make with \( {\overrightarrow Q } \) is , ..............

Hint:

Use \(\vec R=\vec P+\vec Q\) with \(|\vec R|=|\vec P|\) to get a relation between \(\vec P\cdot\vec Q\) and \(Q^2\), then take the dot product of \(2\vec P+\vec Q\) with \(\vec Q\).

Answer 1

Correct Answer: 90

Step 1: \(|\vec R|=|\vec P|\Rightarrow |\vec P+\vec Q|^2=P^2\Rightarrow P^2+Q^2+2\vec P\cdot\vec Q=P^2\Rightarrow Q^2=-2\,\vec P\cdot\vec Q\).
Step 2: New resultant \(\vec R'=2\vec P+\vec Q\). Then \(\vec R'\cdot\vec Q=2\vec P\cdot\vec Q+Q^2=-Q^2+Q^2=0\).
Step 3: A zero dot product means \(\vec R'\) is perpendicular to \(\vec Q\), so \(\theta=90^\circ\).
Answer: 90