Geometric Interpretation Of Complex Number Multiplication Example

Hey guys! Ever wondered how multiplying complex numbers translates into cool geometric transformations on the complex plane? It's way more fascinating than just crunching numbers. Let's dive into a specific example and break it down step by step. We're going to explore how to geometrically determine the product of two complex numbers, $z = 25 - i$ and $w = \sqrt{3} + 3i$. This involves understanding how the magnitude and argument (angle) of complex numbers play a role in multiplication. So, grab your thinking caps, and let's get started!

Geometric Interpretation of Complex Number Multiplication

When we talk about geometrically determining the product of complex numbers, we're essentially visualizing what happens when we multiply them on the complex plane. Think of the complex plane as a map where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. Each complex number can be plotted as a point on this plane, and it can also be thought of as a vector originating from the origin (0,0) to that point. The key to understanding complex number multiplication geometrically lies in two concepts: magnitude and argument. The magnitude of a complex number is its distance from the origin, which we can calculate using the Pythagorean theorem. The argument is the angle that the vector makes with the positive real axis, measured counterclockwise.

When you multiply two complex numbers, the magnitude of the resulting complex number is the product of the magnitudes of the original numbers. This means that if you have two complex numbers, let's say $z$ and $w$, the magnitude of their product $z*w$ will be the magnitude of $z$ multiplied by the magnitude of $w$. So, if $z$ has a magnitude of 5 and $w$ has a magnitude of 3, then their product will have a magnitude of 15. This corresponds to stretching or shrinking the vector representing $z$ by a factor equal to the magnitude of $w$ (or vice versa). For example, if the magnitude of $w$ is greater than 1, the vector representing $z$ is stretched; if it's less than 1, the vector is shrunk. On the other hand, the argument of the product is the sum of the arguments of the original numbers. This is where the rotation comes in. If the argument of $z$ is 30 degrees and the argument of $w$ is 60 degrees, then the argument of their product will be 90 degrees. This means that the vector representing $z$ is rotated counterclockwise by an angle equal to the argument of $w$. Combining these two effects – scaling by the magnitudes and rotating by the arguments – gives us the complete geometric picture of complex number multiplication. It's a powerful way to visualize complex number operations and understand their effects.

Applying the Concept to $z = 25 - i$ and $w = \sqrt{3} + 3i$

Now, let's apply these concepts to our specific example: $z = 25 - i$ and $w = \sqrt{3} + 3i$. Our goal is to figure out how to geometrically obtain the product $z*w$. First, we need to determine the magnitude and argument of each complex number separately. For $z = 25 - i$, the magnitude, denoted as |z|, is calculated as the square root of the sum of the squares of its real and imaginary parts. In this case, |z| = \sqrt{25^2 + (-1)^2} = \sqrt{626}. The argument of $z$, denoted as arg(z), is the angle formed by the vector representing $z$ with the positive real axis. Since $z$ has a positive real part (25) and a negative imaginary part (-1), it lies in the fourth quadrant. We can find the reference angle (the acute angle formed with the x-axis) using the arctangent function: arctan(|-1|/25). The actual argument will be this angle subtracted from 360 degrees (or -arctan(|-1|/25) in radians).

Next, let's consider $w = \sqrt{3} + 3i$. The magnitude of $w$, |w|, is \sqrt(\sqrt{3})^2 + 3^2} = \sqrt{3 + 9} = \sqrt{12} = 2\sqrt{3}. The argument of $w$, arg(w), is the angle formed by the vector representing $w$ with the positive real axis. Since both the real part (\sqrt{3}) and the imaginary part (3) are positive, $w$ lies in the first quadrant. We can find the argument using the arctangent function arg(w) = arctan(3/\sqrt{3) = arctan(\sqrt{3}). The angle whose tangent is \sqrt{3} is 60 degrees (or \pi/3 radians). Now we have all the pieces we need to describe the geometric transformation. Multiplying $z$ by $w$ will result in a complex number whose magnitude is |z| * |w| = \sqrt{626} * 2\sqrt{3} = 2\sqrt{1878}, and whose argument is arg(z) + arg(w). Geometrically, this means that the vector representing $z$ will be stretched by a factor of $2\sqrt{3}$ (the magnitude of $w$) and rotated counterclockwise by 60 degrees (the argument of $w$). So, by understanding the magnitudes and arguments, we can visualize how complex number multiplication transforms vectors on the complex plane.

Analyzing the Given Options

Now that we've thoroughly discussed the geometric interpretation of complex number multiplication and applied it to our specific example, $z = 25 - i$ and $w = \sqrt{3} + 3i$, we're well-equipped to analyze the given options and determine which one accurately describes the geometric transformation involved in finding the product of $z$ and $w$. Remember, multiplying complex numbers involves two key geometric operations: scaling (stretching or shrinking) and rotation. The scaling factor is determined by the magnitude of the complex number we're multiplying by (in this case, $w$), and the rotation angle is determined by the argument of that complex number. Let's look at the options one by one, keeping in mind what we've learned about magnitudes and arguments.

Option A states: "Stretch $z$ by a factor of $\sqrt{3}$ and rotate $30^{\circ}$ counterclockwise." To evaluate this option, we need to compare it with what we've already calculated for $w = \sqrt{3} + 3i$. We found that the magnitude of $w$ is $2\sqrt{3}$, not $\sqrt{3}$, and the argument of $w$ is 60 degrees, not 30 degrees. Therefore, this option is incorrect because it doesn't accurately reflect the scaling and rotation caused by multiplying by $w$. The stretching factor is wrong, and the rotation angle is also incorrect.

Option B (which wasn't fully provided) likely contains a different stretching factor and rotation angle. To determine the correct answer, we need an option that accurately reflects a stretch by a factor of $2\sqrt{3}$ (the magnitude of $w$) and a counterclockwise rotation of 60 degrees (the argument of $w$). Without the full list of options, we can confidently say that Option A is incorrect. The correct statement will accurately describe the scaling and rotation that occur when $z$ is multiplied by $w$. Remember, the magnitude of $w$ determines the stretch factor, and the argument of $w$ determines the rotation angle. So, look for the option that mentions stretching by $2\sqrt{3}$ and rotating by 60 degrees. Understanding these geometric transformations is crucial for working with complex numbers effectively. It provides a visual way to interpret complex number operations, making them more intuitive and easier to grasp.

The Correct Geometric Transformation

To reiterate, when we multiply a complex number $z$ by another complex number $w$, we're performing a geometric transformation on $z$ in the complex plane. This transformation involves two main components: scaling and rotation. The scaling factor is determined by the magnitude of $w$, denoted as |w|, and the rotation is determined by the argument of $w$, denoted as arg(w). In our specific case, we have $z = 25 - i$ and $w = \sqrt{3} + 3i$. We've already calculated that the magnitude of $w$ is |w| = $2\sqrt{3}$, and the argument of $w$ is arg(w) = 60 degrees (or \pi/3 radians). This means that when we multiply $z$ by $w$, the vector representing $z$ in the complex plane will be stretched by a factor of $2\sqrt{3}$ and rotated counterclockwise by 60 degrees. This is the fundamental principle behind geometrically determining the product of complex numbers.

The magnitude of the product $z*w$ is the product of the magnitudes of $z$ and $w$, and the argument of the product $z*w$ is the sum of the arguments of $z$ and $w$. By understanding this, we can visualize the transformation that occurs when we multiply complex numbers. Think of it like this: $w$ acts like a geometric operator on $z$, scaling its length and rotating its direction. This is a powerful way to understand complex number multiplication, and it has applications in various fields, including physics, engineering, and computer graphics. For example, in signal processing, complex numbers are used to represent signals, and multiplication corresponds to filtering or modulation operations. In computer graphics, complex numbers can be used to represent rotations and scaling in 2D space. The ability to visualize these operations geometrically makes it easier to understand and apply complex number arithmetic.

So, to definitively answer the question, we need an option that explicitly states: Stretch $z$ by a factor of $2\sqrt{3}$ and rotate 60 degrees counterclockwise. This statement accurately captures the geometric effect of multiplying $z$ by $w$. Remember, the magnitude of $w$ dictates the scaling, and the argument of $w$ dictates the rotation. Understanding these principles will help you tackle similar problems and deepen your understanding of complex numbers. Keep practicing, and you'll become a pro at visualizing complex number operations!

Original Question: Which statement describes how to geometrically determine the product of $z=25-i$ and $w=\sqrt{3}+3 i$ on the complex plane?

Repaired Question: How can we geometrically describe the product of the complex numbers $z = 25 - i$ and $w = \sqrt{3} + 3i$ on the complex plane?

Geometric Interpretation of Complex Number Multiplication Example