Question

Three vectors \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\), and \(\overrightarrow{OR}\) each of magnitude \(A\) are acting as shown in figure. The resultant of the three vectors is \(A \sqrt{x}\). The value of \(x\) is ______.

Answer 1

Correct Answer: 3

From the given diagram: Vectors $\overrightarrow{OP}$, $\overrightarrow{OQ}$, and $\overrightarrow{OR}$ form angles of $90^\circ$, $45^\circ$, and so on.
The resultant of the three vectors is:
\[\overrightarrow{R} = \overrightarrow{OP} + \overrightarrow{OQ} + \overrightarrow{OR}.\]
The magnitude is:
\[|\overrightarrow{R}| = \sqrt{\left(A + \frac{A}{\sqrt{2}}\right)^2 + \left(A + \frac{A}{\sqrt{2}}\right)^2}.\]
\[|\overrightarrow{R}| = \sqrt{\left(A + \frac{A}{\sqrt{2}}\right)^2 + \left(\frac{A}{\sqrt{2}}\right)^2}.\]
Simplify:
\[|\overrightarrow{R}| = A\sqrt{3}.\]
Thus, $x = 3$.
Final Answer: $x = 3$.