The points corresponding to the four complex numbers given by
`z_1 = 2 text(cis) (pi/3),\ z_2 = text(cis) ((3 pi)/4),\ z_3 = 2 text(cis)(-(2 pi)/3),\ z_4 = text(cis) (-pi/4)`
are the vertices of a parallelogram in the complex plane.
Which one of the following statements is not true?
A. The acute angle between the diagonals of the parallelogram is `(5 pi)/12`
B. The diagonals of the parallelogram have lengths 2 and 4
C. `z_1 z_2 z_3 z_4 = 0 `
D. `z_1 + z_2 + z_3 + z_4 = 0`
E. `1 <= |z| <= 2` for all four of `z_1, z_2, z_3, z_4`
Show Worked Solution
`text(Consider each option:)`
| `A: qquad theta_1` | `= pi/3 + pi/4=(7 pi)/12` |
| `theta_2` | `= pi – (7 pi)/12= (5 pi)/12\ \ text{(correct)}` |
| `B: qquad l_1` | `= 2 + 2 = 4` |
| `l_2` | `= 1+ 1 = 2\ \ text{(correct)}` |
`C: qquad 2 text(cis) (pi/3) ⋅ text(cis) ((3 pi)/4) ⋅ 2 text(cis) (-(2 pi)/3) ⋅ text(cis) (-pi/4)`
`!= 0\ \ text{(by CAS → incorrect)}`
`=> C`