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

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: $x = 3$.

Answer 2

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: $x = 3$.