Geometric Representation of Complex Number

Let $z=2+2\sqrt{3}i$ then plot the complex numbers $iz,\omega z,$ and ${\omega }^{2}z$ on the complex plane, where $\omega$ is a cube root of unity.

Let $z=1+\sqrt{3}i$ then plot the complex numbers $iz,\omega z,$ and ${\omega }^{2}z$ on the complex plane, where $\omega$ is a cube root of unity.

Given that a complex number $z_1=2+2i$ is rotated by $\frac{\mathrm{\pi }}{6}$ radians in an anti-clockwise direction in the complex plane in order to get a new complex number denoted by $z_2$. Hence, determine the complex number $z_2$.

Given that a complex number $z_1=1+i$ is rotated by $\frac{\mathrm{\pi }}{6}$ radians in an anti-clockwise direction in the complex plane in order to get a new complex number denoted by $z_2$. Hence, determine the complex number $z_2$.

Verify whether the points $\left(1+3i\right)$,$\left(3-6i\right)$,$\left(3+6i\right)$ and $\left(2+3i\right)$​ in the complex plane are the vertices of a square or not

Verify whether the points $\left(4+3i\right)$,$\left(3-4i\right)$,$\left(1-i\right)$ and $\left(2+3i\right)$​ in the complex plane are the vertices of a square or not

Verify whether the points $\left(1+3i\right)$,$\left(1-3i\right)$,$\left(1-i\right)$ and $\left(1+i\right)$​ in the complex plane are the vertices of a square or not

Verify whether the points $\left(2+4i\right)$,$\left(2-4i\right)$,$\left(1-i\right)$ and $\left(1+i\right)$​ in the complex plane are the vertices of a square or not

Verify whether the points $\left(3+3i\right)$,$\left(3-4i\right)$,$\left(2-4i\right)$ and $\left(2+5i\right)$​ in the complex plane are the vertices of a rhombus or not

Verify whether the points $\left(3+3i\right)$,$\left(3-4i\right)$,$\left(2-4i\right)$ and $\left(3+3i\right)$​ in the complex plane are the vertices of a rhombus or not

Verify whether the points $\left(2+3i\right)$,$\left(3-4i\right)$,$\left(2-2i\right)$ and $\left(3+3i\right)$​ in the complex plane are the vertices of a rhombus or not

Verify whether the points $\left(2+3i\right)$,$\left(3-4i\right)$,$\left(1-i\right)$ and $\left(3+3i\right)$​ in the complex plane are the vertices of a rhombus or not

Verify whether the points $\left(3+4i\right)$,$\left(3+8i\right)$,$\left(5+2i\right)$ and $\left(3+3i\right)$​ in the complex plane are the vertices of a parallelogram or not

Verify whether the points $\left(3+4i\right)$,$\left(3-4i\right)$,$\left(5+2i\right)$ and $\left(3+3i\right)$​ in the complex plane are the vertices of a parallelogram or not

Verify whether the points $\left(2+4i\right)$,$\left(3-4i\right)$,$\left(5+2i\right)$ and $\left(3+2i\right)$​ in the complex plane are the vertices of a parallelogram or not

Verify whether the points $\left(2+3i\right)$,$\left(3-4i\right)$,$\left(1+2i\right)$ and $\left(3+2i\right)$​ in the complex plane are the vertices of a parallelogram or not

Let $A\left(Z_1\right), B\left(Z_2\right) \mathrm{and} C\left(Z\right)$ be the three points on a line. If C divides line AB (internally) in ratio of $4:5$ and $Z_1=3x+2iy$, $Z_2=6x+7iy$

Find $C\left(Z\right)$.

Let $A\left(Z1\right), B\left(Z2\right) \mathrm{and} C\left(Z\right)$ be the three points on a line. If C divides line AB (internally) in ratio of $5:4$ and $Z_1=3x+2iy$, $Z_2=6x+7iy$

Find $C\left(Z\right)$.