Let's solve the problem step by step: ### **Given:** - Magnitude of \( \vec{OA}, \vec{OB}, \vec{OC} \) are equal. - Find the direction of \( \vec{OA} + \vec{OB} - \vec{OC} \) with the x-axis. Let the magnitude of each vector = \( r \) (let's take \( r = 1 \) for simplicity). --- ### **Step 1: Write the Vectors in Component Form** From the diagram: - \( \vec{OA} \) makes a 30° angle with x-axis (in first quadrant). - \( \vec{OB} \) makes a 60° angle with negative x-axis (in fourth quadrant below x-axis). - \( \vec{OC} \) makes a 45° angle with positive y-axis (second quadrant). Let’s break them into components: #### **\( \vec{OA} \)** Angle with x-axis = 30° \[ \vec{OA} = r(\cos 30^\circ \, \hat{i} + \sin 30^\circ \, \hat{j}) \] \[ = (\frac{\sqrt{3}}{2} \hat{i} + \frac{1}{2} \hat{j}) \] #### **\( \vec{OB} \)** Angle with x-axis = 60°, but it's in the fourth quadrant (so, x positive, y negative). \[ \vec{OB} = r(\cos 60^\circ \, \hat{i} - \sin 60^\circ \, \hat{j}) \] \[ = (\frac{1}{2} \hat{i} - \frac{\sqrt{3}}{2} \hat{j}) \] #### **\( \vec{OC} \)** Angle with x-axis = 45°, but it's in the second quadrant (so, x negative, y positive). \[ \vec{OC} = r(-\cos 45^\circ \, \hat{i} + \sin 45^\circ \, \hat{j}) \] \[ = (-\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j}) \] --- ### **Step 2: Compute \( \vec{OA} + \vec{OB} - \vec{OC} \)** Sum the i and j components: #### **i-component:** \[ \left(\frac{\sqrt{3}}{2} + \frac{1}{2} - \left(-\frac{1}{\sqrt{2}}\right)\right) = \frac{\sqrt{3}}{2} + \frac{1}{2} + \frac{1}{\sqrt{2}} \] #### **j-component:** \[ \left(\frac{1}{2} - \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}}\right) = \frac{1}{2} - \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \] --- ### **Step 3: Direction with x-axis** Let \( \theta \) be the angle with the x-axis: \[ \tan\theta = \frac{\text{y-component}}{\text{x-component}} = \frac{\frac{1}{2} - \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2} + \frac{1}{2} + \frac{1}{\sqrt{2}}} \] Let's simplify numerator and denominator: #### **Numerator:** \[ \frac{1}{2} - \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \] #### **Denominator:** \[ \frac{\sqrt{3}}{2} + \frac{1}{2} + \frac{1}{\sqrt{2}} \] Multiply numerator and denominator by 2 to clear denominators: Numerator: \[ 1 - \sqrt{3} - \sqrt{2} \] Denominator: \[ \sqrt{3} + 1 + \sqrt{2} \] Now, match with the options: - Option (a): \(\tan^{-1}\left(\frac{1-\sqrt{3}-\sqrt{2}}{1+\sqrt{3}+\sqrt{2}}\right)\) **This matches exactly!** --- ## **Final Answer** \[ \boxed{ \text{Option (a): } \tan^{-1}\left(\frac{1-\sqrt{3}-\sqrt{2}}{1+\sqrt{3}+\sqrt{2}}\right) } \] Let me know if you need any further explanation!