Complex Numbers
Rotation of Complex Numbers — Principal Argument
nta_pyq_2023_apr
Grade 11
Question:
Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the point obtained by the rotation of $z_2=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1-w_2$ is equal to
$\pi-\tan^{-1}\dfrac{8}{9}$
$-\pi+\tan^{-1}\dfrac{33}{5}$
$-\pi+\tan^{-1}\dfrac{8}{9}$
$\pi-\tan^{-1}\dfrac{33}{5}$
Step-by-Step Solution
Key Concept: Multiply by $i$ for anticlockwise $90°$: $w_1=i\cdot z_1=i(5+4i)=-4+5i$. Multiply by $-i$ for clockwise $90°$: $w_2=-iz_2=-i(3+5i)=5-3i$.
$w_1=-4+5i$, $w_2=5-3i$. $w_1-w_2=-9+8i$. Argument $=\pi-\tan^{-1}\frac{8}{9}$.
Correct Answer: 1