Geometric Interpretation Of Complex Number Multiplication Let $z = 3 + 2i$ And $w = 4 + I$

Introduction

In the realm of complex numbers, multiplication unveils a fascinating geometric interpretation. When we multiply two complex numbers, $z$ and $w$, the resulting complex number's magnitude is the product of the magnitudes of $z$ and $w$, and its argument (angle with the positive real axis) is the sum of the arguments of $z$ and $w$. This article delves into the geometric construction of the product $z \times w$ on the complex plane, where $z = 3 + 2i$ and $w = 4 + i$. Understanding this geometric interpretation provides a powerful visual tool for grasping complex number arithmetic.

Complex Numbers as Vectors

To understand the geometric construction, it's crucial to visualize complex numbers as vectors in the complex plane. A complex number $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($\sqrt{-1}$), can be represented as a vector from the origin to the point $(a, b)$ in the complex plane. The real part, $a$, corresponds to the x-coordinate, and the imaginary part, $b$, corresponds to the y-coordinate. The magnitude or modulus of $z$, denoted as $|z|$, is the length of this vector, calculated using the Pythagorean theorem: $|z| = \sqrt{a^2 + b^2}$. The argument of $z$, denoted as $\text{arg}(z)$, is the angle this vector makes with the positive real axis, measured counterclockwise. It can be found using trigonometric functions, specifically the arctangent function: $\text{arg}(z) = \arctan(\frac{b}{a})$. However, care must be taken to choose the correct quadrant for the angle based on the signs of $a$ and $b$.

For our given complex numbers, $z = 3 + 2i$ and $w = 4 + i$, we can visualize them as vectors. The vector for $z$ extends from the origin to the point $(3, 2)$, and the vector for $w$ extends from the origin to the point $(4, 1)$. The magnitudes and arguments of these vectors play a crucial role in understanding their geometric construction.

Multiplication in Polar Form

The geometric interpretation of complex number multiplication becomes particularly clear when complex numbers are expressed in polar form. The polar form of a complex number $z = a + bi$ is given by $z = r(\cos \theta + i \sin \theta)$, where $r = |z|$ is the magnitude and $\theta = \text{arg}(z)$ is the argument. This representation highlights the magnitude and direction of the complex number as a vector. When we multiply two complex numbers in polar form, the magnitudes multiply, and the arguments add.

Let $z = r_1(\cos \theta_1 + i \sin \theta_1)$ and $w = r_2(\cos \theta_2 + i \sin \theta_2)$. Then their product, $zw$, is given by:

$zw = r_1r_2 [(\cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2) + i(\sin \theta_1 \cos \theta_2 + \cos \theta_1 \sin \theta_2)]$

Using trigonometric identities, this can be simplified to:

$zw = r_1r_2[\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$

This equation clearly shows that the magnitude of the product, $|zw|$, is the product of the magnitudes, $r_1r_2$, and the argument of the product, $\text{arg}(zw)$, is the sum of the arguments, $\theta_1 + \theta_2$. Geometrically, this means that multiplying $z$ by $w$ stretches the vector representing $z$ by a factor of $|w|$ and rotates it counterclockwise by the angle $\text{arg}(w)$. This stretching and rotation is the key to understanding the geometric construction of complex number multiplication.

Geometric Construction of $z \times w$

Now, let's apply this understanding to our specific complex numbers, $z = 3 + 2i$ and $w = 4 + i$. First, we need to find the magnitudes and arguments of $z$ and $w$.

For $z = 3 + 2i$:

• Magnitude: $|z| = \sqrt{3^2 + 2^2} = \sqrt{13}$
• Argument: $\text{arg}(z) = \arctan(\frac{2}{3}) \approx 33.69^{\circ}$

For $w = 4 + i$:

• Magnitude: $|w| = \sqrt{4^2 + 1^2} = \sqrt{17}$
• Argument: $\text{arg}(w) = \arctan(\frac{1}{4}) \approx 14.04^{\circ}$

To find the product $zw$, we multiply the magnitudes and add the arguments:

• Magnitude of $zw$: $|zw| = |z| \cdot |w| = \sqrt{13} \cdot \sqrt{17} = \sqrt{221}$
• Argument of $zw$: $\text{arg}(zw) = \text{arg}(z) + \text{arg}(w) \approx 33.69^{\circ} + 14.04^{\circ} \approx 47.73^{\circ}$

Geometrically, this means that the vector representing $z$ is stretched by a factor of $\sqrt{17}$ (the magnitude of $w$) and rotated counterclockwise by approximately $14.04^{\circ}$ (the argument of $w$). The resulting vector represents the complex number $zw$.

To find the rectangular form of $zw$, we can multiply $z$ and $w$ directly:

$zw = (3 + 2i)(4 + i) = 3(4) + 3(i) + 2i(4) + 2i(i) = 12 + 3i + 8i + 2i^2 = 12 + 11i - 2 = 10 + 11i$

The magnitude of $10 + 11i$ is $\sqrt{10^2 + 11^2} = \sqrt{100 + 121} = \sqrt{221}$, which agrees with our previous calculation. The argument is $\arctan(\frac{11}{10}) \approx 47.73^{\circ}$, also confirming our geometric interpretation.

Visualizing the Transformation

Imagine the vector representing $z$ in the complex plane. Multiplying by $w$ can be visualized as two distinct operations:

1. Stretching: The vector $z$ is stretched by a factor equal to the magnitude of $w$, which is $\sqrt{17}$. This means the length of the vector $z$ is multiplied by $\sqrt{17}$.
2. Rotation: The stretched vector is then rotated counterclockwise by an angle equal to the argument of $w$, which is approximately $14.04^{\circ}$. This rotation changes the direction of the vector.

The final vector obtained after these two operations represents the product $zw$ in the complex plane. Its length is $\sqrt{221}$, and its angle with the positive real axis is approximately $47.73^{\circ}$.

Conclusion

The geometric construction of the product $z \times w$ on the complex plane vividly demonstrates the effect of complex number multiplication. When we multiply two complex numbers, the magnitude of one complex number stretches the other by a factor equal to its magnitude, and the argument of one complex number rotates the other by its angle. For $z = 3 + 2i$ and $w = 4 + i$, this means that $z$ is stretched by a factor of $\sqrt{17}$ and rotated approximately $14.04^{\circ}$ counterclockwise. This geometric interpretation provides a powerful visual tool for understanding complex number operations and their impact on the complex plane.