2022 SJI P1 Q5

The complex numbers \(u\) and \(w\) are defined by \(u=-1+7\mathrm{i}\) and \(w=3+4\mathrm{i}\), where \(\mathrm{i}^2=-1\).

1. Find \(u-2w\) and \(\dfrac uw\) in the form \(x+\mathrm{i}y\), where \(x\) and \(y\) are real numbers.[3]

Points \(A\) and \(B\) represent the complex numbers \(u\) and \(w\) respectively on an Argand diagram with \(O\) as the origin.

2. Given that angle \(AOB=\arg\left(\dfrac uw\right)\), find angle \(AOB\).[2]
3. Hence, show that \(\arctan\left(\dfrac43\right)+\arctan7=\dfrac{3\pi}4\).[4]