Suppose ω is a cube root of unity with ω ≠ 1. Suppose P and Q are the points on the complex plane defined by ω and ω 2. If O is the origin, then what is the angle between OP and OQ?
120°
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
| Complex Number | Point on Complex Plane | Argument (Angle from positive x-axis) |
|---|---|---|
| ω | P | \(120^\circ\) (\(2\pi/3\) rad) |
| ω2 | Q | \(240^\circ\) (\(4\pi/3\) rad) or \(-120^\circ\) (\(-2\pi/3\) rad) |
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
| Property | Description | Formula/Value |
|---|---|---|
| Equation | Defining equation for cube roots of unity | \(z^3 = 1\) |
| Roots | The three cube roots of unity | \(1, \omega, \omega^2\) |
| Sum of Roots | Sum of the three cube roots | \(1 + \omega + \omega^2 = 0\) |
| Product of Roots | Product of the three cube roots | \(1 \cdot \omega \cdot \omega^2 = \omega^3 = 1\) |
| Arguments | Angles of the roots on the complex plane | \(0^\circ, 120^\circ, 240^\circ\) |
| Modulus | Magnitude of each root | \(|1|=1, |\omega|=1, |\omega^2|=1\) |
| Relationship | Relationship between non-real roots | \(\omega^2 = \bar{\omega}\) and \(\omega = \bar{\omega}^2\) |
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 \(240^\circ - 120^\circ = 120^\circ\).
Important Questions from Complex Numbers
Which one of the following is a square root of \(-\sqrt{-1} \)?
What are the roots of equation-I ?
Which one of the following is a root of equation-II?
What is the number of common roots of equation-I and equation-II?
If \(z=\frac{1+i √{3}}{1-i √{3}}\) where i = √-1 then what is the argument of z ?
