Understanding Cube Roots of Unity on the Complex Plane

The question asks for the angle between two line segments, OP and OQ, where O is the origin, and P and Q represent the complex numbers ω and ω², respectively, with ω being a non-real cube root of unity.

The cube roots of unity are the solutions to the equation \(z^3 = 1\). These solutions are \(1\), \(\omega\), and \(\omega^2\). Since ω ≠ 1, the other two roots are the non-real ones.

In the complex plane, these roots are located on the unit circle centered at the origin. They divide the circle into three equal parts, meaning the angle between consecutive roots (from the origin) is \(360^\circ / 3 = 120^\circ\).

Locating Points P and Q

Let's represent the cube roots of unity in polar form. The modulus of each root is 1.

• The root 1 corresponds to an angle of \(0^\circ\) with the positive real axis.
• The non-real roots ω and ω² correspond to angles of \(120^\circ\) and \(240^\circ\) (or \(-120^\circ\)) with the positive real axis, respectively.

In terms of complex numbers:

• \(1 = 1 \cdot (\cos(0^\circ) + i \sin(0^\circ))\)
• \(\omega = 1 \cdot (\cos(120^\circ) + i \sin(120^\circ))\) or \(e^{i 2\pi/3}\)
• \(\omega^2 = 1 \cdot (\cos(240^\circ) + i \sin(240^\circ))\) or \(e^{i 4\pi/3}\)

Point P is defined by the complex number ω. The line segment OP is a vector from the origin O to the point P, which lies on the unit circle at an angle of \(120^\circ\) from the positive real axis.

Point Q is defined by the complex number ω². The line segment OQ is a vector from the origin O to the point Q, which lies on the unit circle at an angle of \(240^\circ\) from the positive real axis.

Determining the Angle Between OP and OQ using Arguments

The angle of a complex number \(z\) from the positive x-axis is given by its argument, \(\text{arg}(z)\). For points P and Q represented by ω and ω², the angle between OP and OQ is the difference between the arguments of ω² and ω.

The argument of ω, \(\text{arg}(\omega)\), is \(120^\circ\) (or \(2\pi/3\) radians).

The argument of ω², \(\text{arg}(\omega^2)\), is \(240^\circ\) (or \(4\pi/3\) radians).

The angle between OP and OQ is the absolute difference of their arguments:

\(\text{Angle} = |\text{arg}(\omega^2) - \text{arg}(\omega)|\)

Substituting the degree values:

\(\text{Angle} = |240^\circ - 120^\circ| = |120^\circ| = 120^\circ\)

Alternatively, using radians:

\(\text{Angle} = |\frac{4\pi}{3} - \frac{2\pi}{3}| = |\frac{2\pi}{3}| = \frac{2\pi}{3}\)

Converting \(\frac{2\pi}{3}\) radians to degrees:

\(\frac{2\pi}{3} \text{ radians} \times \frac{180^\circ}{\pi \text{ radians}} = \frac{2 \times 180^\circ}{3} = 2 \times 60^\circ = 120^\circ\)

Using Properties of Complex Number Division

The angle between two vectors from the origin representing complex numbers \(z_1\) and \(z_2\) can also be found as the argument of the quotient \(z_2/z_1\).

Here, \(z_1 = \omega\) and \(z_2 = \omega^2\). The angle is \(\text{arg}(\omega^2 / \omega)\).

\(\frac{\omega^2}{\omega} = \omega\)

So, the angle is \(\text{arg}(\omega)\).

Since ω is a non-real cube root of unity, its principal argument is \(120^\circ\) or \(2\pi/3\) radians.

Therefore, the angle between OP and OQ is \(120^\circ\).

Summary of Angle Calculation for Cube Roots of Unity

The angle between OP and OQ is the difference between their arguments: \(240^\circ - 120^\circ = 120^\circ\).

Revision Table: Important Properties of Cube Roots of Unity

Additional Information: Geometric Interpretation of Complex Multiplication

When you multiply two complex numbers in polar form, \(z_1 = r_1 e^{i \theta_1}\) and \(z_2 = r_2 e^{i \theta_2}\), the result is \(z_1 z_2 = (r_1 r_2) e^{i (\theta_1 + \theta_2)}\). This means the modulus of the product is the product of the moduli, and the argument of the product is the sum of the arguments.

In our case, \(\omega^2 = \omega \cdot \omega\).

• The modulus of ω is 1, so the modulus of \(\omega^2\) is \(1 \times 1 = 1\). Both points P and Q are on the unit circle.
• If \(\text{arg}(\omega) = 120^\circ\), then \(\text{arg}(\omega^2) = \text{arg}(\omega) + \text{arg}(\omega) = 120^\circ + 120^\circ = 240^\circ\).

The vectors OP and OQ originate from the same point (the origin). The angle between them is the difference in their orientations relative to the positive x-axis. Since OP is at \(120^\circ\) and OQ is at \(240^\circ\), the angle between them is \(240^\circ - 120^\circ = 120^\circ\).