Let w_1 be the point obtained by the rotation of z_1=5+4 i about the origin through a righ…

Question

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 :

Accepted Answer

To solve the problem, let's break it down step by step.

Find $w_{1}: 1 : Find $w\_{1}$

Given $z\_{1} = 5 + 4i$.

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 $z\_{1}$.

Therefore, $w\_{1}$ becomes :

$w\_{1} = -4 + 5i$

Find $w\_{2}: 2 : Find $w_{2}$

Given $z_{2} = 3 + 5i$.

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 $z_{2}$.

Therefore, $w_{2}$ becomes :

$w_{2} = 5 - 3i$

Calculate $w_{1} - w_{2}: 3 : Calculate $w\_{1} - w\_{2}$

$w\_{1} - w\_{2} = (-4 + 5i) - (5 - 3i)$

$w\_{1} - w\_{2} = -9 + 8i$

Find the principal argumen: 4 : Find the principal argument

To find the principal argument, we need to compute the tangent inverse of the ratio of the imaginary part to the real part.

Argument = $	an\^{-1}\left(\frac{	ext{Imaginary part}}{	ext{Real part}}ight)$

Argument = $	an\^{-1}\left(\frac{8}{-9}ight)$

Argument = $-	an\^{-1}\left(\frac{8}{9}ight)$

Since it's in the third quadrant, the principal argument is :

Argument = $\pi + (-	an\^{-1}\left(\frac{8}{9}ight))$

Argument = $\pi - 	an\^{-1}\left(\frac{8}{9}ight)$

So, the correct option is :

Option C : $

Solutions · 1