If area of triangle formed by z, ωz and z + ωz is 16√3square units (where ω is complex cube root of unity) the |z| is equal to :-Correct answer is '8'. Can you explain this answer?

Question

If area of triangle formed by z, &omega;z and z + &omega;z is 16&radic;3square units (where &omega; is complex cube root of unity) the |z| is equal to :-Correct answer is '8'. Can you explain this answer?

Answer

To find the area of a triangle formed by complex numbers, we can use the formula:

Area = (1/2) * |Im((z_2 - z_1) * (z_3 - z_1))|

Here, z_1, z_2, and z_3 are the complex numbers that form the vertices of the triangle.

Let's say the complex numbers are z_1, z_2, and z_3.

Then, the area of the triangle formed by z_1, z_2, and z_3 can be calculated as:

Area = (1/2) * |Im((z_2 - z_1) * (z_3 - z_1))|

This formula calculates the imaginary part of the product of the differences between the complex numbers.

Note: The absolute value of the imaginary part is taken because the area of a triangle is always positive.

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