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Physics · MOTION IN A PLANE
हिन्दीVector \( \mathbf{P} \) has magnitude 6 units and points along the positive x-axis. Vector \( \mathbf{Q} \) has magnitude 10 units and makes an angle of \( 120^\circ \) with the positive x-axis. What is the unit vector along the direction of the resultant vector \( \mathbf{R} = \mathbf{P} + \mathbf{Q} \)?
Correct answer · Option 4
\( \frac{\hat{\mathbf{i}} + 5\sqrt{3}\hat{\mathbf{j}}}{2\sqrt{19}} \)
Explanation
\( \mathbf{P} = 6\hat{\mathbf{i}} \) and \( \mathbf{Q} = 10\cos 120^\circ \hat{\mathbf{i}} + 10\sin 120^\circ \hat{\mathbf{j}} = -5\hat{\mathbf{i}} + 5\sqrt{3}\hat{\mathbf{j}} \). Thus \( \mathbf{R} = \hat{\mathbf{i}} + 5\sqrt{3}\hat{\mathbf{j}} \). Its magnitude is \( |\mathbf{R}| = \sqrt{1 + 75} = \sqrt{76} = 2\sqrt{19} \), so the unit vector is \( \frac{\hat{\mathbf{i}} + 5\sqrt{3}\hat{\mathbf{j}}}{2\sqrt{19}} \).
