Geometric Transformation Of Complex Number Product

Let's explore how multiplying complex numbers affects their geometric representation on the coordinate plane. Specifically, we'll examine the complex numbers z = 13 + 13i and w = √6 + i√2, and determine the geometric transformation that results from their product.

Understanding Complex Numbers and Geometric Transformations

Complex numbers, often expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1), can be visualized as points on a complex plane. The horizontal axis represents the real part (a), and the vertical axis represents the imaginary part (b). Multiplying complex numbers has a fascinating geometric interpretation: it involves both scaling (magnification or reduction) and rotation.

To fully grasp the geometric transformation, we'll convert the complex numbers into their polar forms. This representation expresses a complex number in terms of its magnitude (distance from the origin) and its argument (angle with respect to the positive real axis). The polar form simplifies the multiplication process and reveals the scaling and rotation effects more clearly.

Converting to Polar Form

The polar form of a complex number z = a + bi is given by r(cos θ + i sin θ), where r is the magnitude and θ is the argument.

• Magnitude (r): The magnitude r is calculated as r = √(a² + b²), representing the distance of the complex number from the origin in the complex plane.
• Argument (θ): The argument θ is the angle between the positive real axis and the line connecting the origin to the complex number. It can be found using θ = arctan(b/a), taking into account the quadrant in which the complex number lies.

Analyzing z = 13 + 13i

Let's find the polar form of z = 13 + 13i.

• Magnitude of z: r_z = √(13² + 13²) = √(169 + 169) = √(338) = 13√2

• Argument of z: θ_z = arctan(13/13) = arctan(1) = 45° (since 13 + 13i lies in the first quadrant)

Therefore, z = 13√2 (cos 45° + i sin 45°).

Analyzing w = √6 + i√2

Now, let's find the polar form of w = √6 + i√2.

• Magnitude of w: r_w = √((√6)² + (√2)²) = √(6 + 2) = √8 = 2√2

• Argument of w: θ_w = arctan(√2 / √6) = arctan(1/√3) = 30° (since √6 + i√2 lies in the first quadrant)

Therefore, w = 2√2 (cos 30° + i sin 30°).

Multiplying Complex Numbers in Polar Form

When multiplying complex numbers in polar form, the magnitudes are multiplied, and the arguments are added.

If z = r_z (cos θ_z + i sin θ_z) and w = r_w (cos θ_w + i sin θ_w), then:

z w = r_z r_w [cos(θ_z + θ_w) + i sin(θ_z + θ_w)]

Calculating the Product z * w

Let's calculate the product z w using the polar forms we found.

z w = (13√2) * (2√2) [cos(45° + 30°) + i sin(45° + 30°)]

z w = 52 [cos(75°) + i sin(75°)]

Interpreting the Geometric Transformation

The product z w has a magnitude of 52 and an argument of 75°. This tells us that the geometric transformation involves:

1. Scaling: The original complex number z is scaled by a factor of 2√2, which is the magnitude of w.
2. Rotation: The original complex number z is rotated counterclockwise by an angle of 30°, which is the argument of w.

Determining the Scaling Factor

Since the magnitude of z is 13√2 and the magnitude of the product z w is 52, the scaling factor applied to z is:

Scaling Factor = |z w| / |z| = 52 / (13√2) = 4/√2 = 2√2

Summary of the Geometric Transformation

Multiplying z = 13 + 13i by w = √6 + i√2 results in a geometric transformation that scales z by a factor of 2√2 and rotates it counterclockwise by 30°.

• Scaling Factor: 2√2
• Rotation Angle: 30° counterclockwise

Conclusion

The geometric interpretation of multiplying complex numbers provides valuable insights into how these operations affect their representation on the complex plane. By converting complex numbers into polar form, we can easily determine the scaling and rotation effects of multiplication. In this specific case, multiplying z = 13 + 13i by w = √6 + i√2 scales z by a factor of 2√2 and rotates it 30° counterclockwise. Understanding these transformations is crucial in various fields, including physics, engineering, and computer graphics, where complex numbers are frequently used to model and manipulate geometric objects.

For further information on complex numbers and their geometric transformations, you might find the resources at Khan Academy's Complex Numbers helpful.