Let $w_{1}$ be the point obtained by the rotation of $z_{1}=5+4 i$ about the origin through a right angle in the anticlockwise direction, and $w_{2}$ be the point obtained by the rotation of $z_{2}=3+5 i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_{1}-w_{2}$ is equal to :

<p>To solve the problem, let&#39;s break it down step by step.</p> <p><strong>Step 1 :</strong> Find $w_{1}$ </p> <p>Given $z_{1} = 5 + 4i$. </p> <p>When you rotate $z_{1}$ by $90^{\circ}$ anticlockwise about the origin, the real part becomes negative of the imaginary part of $z_{1}$, and the imaginary part becomes the real part of $z_{1}$.</p> <p>Therefore, $w_{1}$ becomes : <br/><br/>$w_{1} = -4 + 5i$</p> <p><strong>Step 2 :</strong> Find $w_{2}$</p> <p>Given $z_{2} = 3 + 5i$.</p> <p>When you rotate $z_{2}$ by $90^{\circ}$ clockwise about the origin, the real part becomes the imaginary part of $z_{2}$, and the imaginary part becomes negative of the real part of $z_{2}$.</p> <p>Therefore, $w_{2}$ becomes : <br/><br/>$w_{2} = 5 - 3i$</p> <p><strong>Step 3 :</strong> Calculate $w_{1} - w_{2}$</p> <p>$w_{1} - w_{2} = (-4 + 5i) - (5 - 3i)$ <br/><br/>$w_{1} - w_{2} = -9 + 8i$</p> <p><strong>Step 4 :</strong> Find the principal argument</p> <p>To find the principal argument, we need to compute the tangent inverse of the ratio of the imaginary part to the real part. <br/><br/>Argument = $\tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right)$ </p> <p>Argument = $\tan^{-1}\left(\frac{8}{-9}\right)$ <br/><br/>Argument = $-\tan^{-1}\left(\frac{8}{9}\right)$ </p> <p>Since it&#39;s in the third quadrant, the principal argument is : <br/><br/>Argument = $\pi + (-\tan^{-1}\left(\frac{8}{9}\right))$ <br/><br/>Argument = $\pi - \tan^{-1}\left(\frac{8}{9}\right)$ </p> <p>So, the correct option is : <br/><br/>Option C : $\pi-\tan^{-1} \frac{8}{9}$</p>