Understand and use addition of complex numbers as vector addition in the complex plane.

Consider the complex numbers \(z_1 = 3 + 2i\) and \(z_2 = -1 + 4i\). a) Determine \(z_1 + z_2\) algebraically. (1 mark) b) Represent \(z_1\), \(z_2\), and \(z_1 + z_2\) as position vectors in the complex plane, and show evidence of the vector addition by sketching these vectors on an Argand diagram. (1 mark) c) Determine the modulus of \(z_1 + z_2\), expressing your answer correct to two decimal places. (1 mark)

Let \(z_1 = 3 + 4i\), \(z_2 = -2 + 5i\), and \(z_3 = 6 - 3i\). Three vectors \(\vec{OA}\), \(\vec{AB}\), and \(\vec{BC}\) represent the complex numbers \(z_1\), \(z_2\), and \(z_3\) respectively in the complex plane, where \(O\) is the origin. Determine whether the quadrilateral \(OABC\) is a parallelogram by using vector addition properties of complex numbers.