2025 RI Promo Q3

The point \(A\) on the Argand diagram below represents the complex number \(w_1\) with modulus \(r\) and argument \(\theta\). It is given that the complex numbers \(w_2\) and \(w_3\) satisfy the equations \(w_2=-w_1\) and \(w_3=iw_2\).

Let \(B\), \(C\) and \(D\) represent the complex numbers \(w_2\), \(w_3\) and \(w_2+w_3\) respectively.

1. On the copy of the Argand diagram in the Printed Answer Book, plot the points \(B\), \(C\) and \(D\), indicating clearly the modulus and argument of \(w_2\) and \(w_3\).[3]
2. State, in radians, the angle \(BOC\).[1]
3. By considering the quadrilateral \(OBDC\), find \(\left|w_2+w_3\right|\) in terms of \(r\) and \(\arg(w_2+w_3)\) in terms of \(\theta\).[2]