Does anyone have a cheat sheet for proving shapes in complex numbers? Whenever I'm given a question like "prove these complex points form a parallelogram" I never know how to prove it and the answer's always smthn random like "the diagonals bisect each other". So is there a method of knowing the proof for each shape?
There is always a purely algebraic method, which is usually awful. There are sometimes purely geometric methods (like for arg(
z -
i) = arg(
z + 1) etc.). If there isn't an obvious purely geometric approach, the efficient answer is likely to involve:
- treating the complex numbers as vectors
- looking for geometric properties that proves the required result
- demonstrating these properties through algebraic representation of vectors
For example... the complex number
z represents a point
A in the first quadrant. If
O is the origin,
B lies in the second quadrant, and
OACB is a square, find the complex number representing point
C. Under what conditions is C located in the second quadrant.
A diagram should make it obvious that side OB is adjacent to side OA in the square.
Properties of a square then dictate that
OB =
i.
OA, and so the complex number
iz represents
B.
Then, using vector reasoning:
Hence, the point C is represented by
z(1 +
i), and so is in the second quadrant if
from the diagram (as
A must be in quadrant 1 and, for
C to be in quadrant 2 given angle
COA is 45 degrees,
OA must be inclined at at least 45 degrees above the real axis), or (algebraically), by solving: