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 ...

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 = ${	an }\^{-1}(\frac{}{})$

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

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

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

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

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

So, the correct option is :

Option C : $

Solutions · 1