Help: Solutions for Ext 2 Cambridge textbook Chapter 1

[tony.li12]

Hey can someone help me with some questions from the Ext2 Cambridge textbook? Chapter 1E Questions 14-17 Thanks!

[fun_jirachi]

Hey there, welcome to the forums

What have you tried (if anything at all)? It's a lot more helpful to you (and us!) if you try the question first so you can target the gaps in your understanding. That being said, here are some hints

Question 14

Geometric approach - draw a parallelogram with sides z and 1 on the Argand plane. Use geometric properties to deduce the result from there.

Question 15

a) \(z_1 - z_2 = z_4 - z_3\) implies that two edges of the quadrilateral are equal and parallel (by definition of vector equality). What does this say about ABCD?
b) Similarly, what does the second condition imply? Don't open spoiler unless you really have to.

Answer

Diagonals bisect - what does this then imply?

Question 16

When you divide two complex numbers, you divide the modulus and subtract the arguments. In vector terms, if the result is of the form \(\cos \theta + i\sin \theta\), then the vectors have equal length and enclose an angle of size \(\theta\).

Question 17

Use the same principle as 16 to prove the result for both angles in question. This is a bit tougher, so I'll provide a full solution if you really need it

Hope this helps