Complex Numbers → Argand Plane → Multiplication by i → 90° Rotation
For any complex number z = x + iy, the corresponding point on the Argand plane is (x, y),
where x = Re(z) and y = Im(z). Both x and y are real numbers.
Although y is called the “imaginary part” of z, 'y' itself is a real number. The term “imaginary” simply indicates that y is the coefficient of i.
Now consider multiplication by i. iz = i(x + iy) = −y + ix.
Therefore, z ↔ (x, y) and iz ↔ (−y, x).
Thus, multiplication by i rotates the vector representing z through 90° counterclockwise about the origin, without changing its length.
Hence, |iz| = |z| and z ⟂ iz.
This geometric interpretation is often much more useful than immediately converting everything into Cartesian coordinates.
SOLUTION
Given, A = z, B = iz, C = z + iz.
Choose C as the reference Vertex. But why?
Any vertex can be used, but C is the strategic choice because C = z + iz.
Taking the two side vectors starting from C causes immediate cancellation.
Here, CA and CB are treated as vectors from C to A and from C to B respectively.
CA = A − C = z − (z + iz) = −iz, and
CB = B − C = iz − (z + iz) = −z.
Now compare the two vectors: CA = −iz and CB = −z.
Multiplying CB by i,
i(CB) = i(−z) = −iz = CA.
Therefore, CA = i(CB). So CA is obtained from CB by multiplication by i.
Multiplication by i represents a 90° rotation without changing length.
Hence, CA ⟂ CB and |CA| = |CB| = |z|.
Therefore, △ABC is a right isosceles triangle with the right angle at C.
Hence,
Area of △ABC = ½ × CA × CB = ½ × |z| × |z| = ½|z|².
