Let w_1 be the point obtained by rotating z_1=5+4i about the origin through a right angle …

Question

Let $w_1$ be the point obtained by rotating $z_1=5+4i$ about the origin through a right angle anticlockwise, and $w_2$ be the point obtained by rotating $z_2=3+5i$ about the origin through a right angle clockwise. Then the principal argument of $w_1-w_2$ is equal to:

Accepted Answer

Rotating anticlockwise by $90^{\circ}$ multiplies by $i$; clockwise by $90^{\circ}$ multiplies by $-i$.

$$w_1=z_1\cdot i=(5+4i)i=-4+5i, \qquad w_2=z_2\cdot(-i)=(3+5i)(-i)=5-3i.$$

$w_1-w_2=(-4-5)+(5+3)i=-9+8i.$

This point lies in the second quadrant, so the principal argument is $\pi-\tan^{-1}\left(\dfrac89\right)$.

Solutions · 1