Calculating Vector Moduli: A Deep Dive Into Complex Numbers

Hey guys! Let's dive into a cool math problem involving complex numbers and vectors. We're given three points, A, B, and C, each represented by a complex number (also known as an affix). Our mission? To calculate the modulus of the vectors OA, OB, and OC. Don't worry, it sounds more complicated than it is. We'll break it down step by step, making sure everyone understands the concepts. This is like a fun treasure hunt, where the treasure is understanding complex numbers and vector magnitudes!

Understanding the Basics: Complex Numbers and Affixes

Okay, before we get our hands dirty with calculations, let's make sure we're all on the same page about complex numbers and their affixes. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (i.e., the square root of -1). The 'a' part is called the real part, and the 'b' part is called the imaginary part. Think of it like this: complex numbers are an extension of real numbers, allowing us to deal with square roots of negative numbers. They're super useful in all sorts of fields, from electrical engineering to quantum physics.

Now, what about the affix? In the context of complex numbers, the affix of a point in the complex plane is simply the complex number that represents that point. So, if we have a point A with the affix z_A = -2 - √3i, it means this point corresponds to the complex number -2 - √3i. Pretty straightforward, right? Imagine the complex plane as a regular coordinate plane (x, y), where the x-axis represents the real part and the y-axis represents the imaginary part. The affix z_A then tells us where point A is located on this plane. The real part (-2) gives the x-coordinate, and the imaginary part (-√3) gives the y-coordinate.

This is where it gets interesting, isn't it? Points A, B, and C each have their own complex affixes: z_A = -2 - √3i, z_B = √14 - √2i, and z_C = -1. Each affix is like a secret code revealing the position of these points in the complex plane. Think of it like a treasure map where the complex affixes are the coordinates of the treasures. Now that we have the coordinates, our goal is to find the distances from the origin (point O) to these points. These distances are the moduli of the vectors OA, OB, and OC. It's like measuring how far each treasure is from the starting point!

To make this clearer, let's think about vectors. A vector is a quantity that has both magnitude (length) and direction. In our case, the vectors OA, OB, and OC start at the origin (0, 0) and end at the points A, B, and C, respectively. The modulus (or absolute value) of a vector is simply its length. So, calculating the modulus of the vectors OA, OB, and OC is the same as finding the lengths of the line segments OA, OB, and OC. Are you still with me? Great! Let's get to the calculations!

Calculating the Modulus of OA

Alright, let's start with the vector OA. We know that the affix of point A is z_A = -2 - √3i. To find the modulus of the vector OA, we need to calculate the distance between the origin (0, 0) and the point A. Remember that the modulus of a complex number z = a + bi is given by the formula |z| = √(a² + b²). It's essentially the Pythagorean theorem in disguise! If you've been wondering, this formula comes straight from the Pythagorean theorem. You know, a² + b² = c²? Well, here, 'a' and 'b' are the real and imaginary parts, and the modulus is the hypotenuse, 'c'.

In our case, for z_A = -2 - √3i, the real part (a) is -2, and the imaginary part (b) is -√3. Applying the formula, we get: |OA| = √((-2)² + (-√3)²) = √(4 + 3) = √7. Therefore, the modulus of the vector OA is √7. It is easy, right? So, the distance from the origin to point A is the square root of 7. It's a nice, neat little number that tells us the length of that vector. We've conquered the first part! We successfully calculated the length of the vector OA using the modulus formula. We essentially found the distance from the origin to the point A in the complex plane. Feel like a champion? You should!

Let's put the result in a more conversational way: so the modulus of the vector OA is √7. This means that if you were to draw a line from the origin to point A on the complex plane, that line would be approximately 2.65 units long. We've effectively measured the length of that line using our understanding of complex numbers and the modulus formula. Pretty neat, huh?

Calculating the Modulus of OB

Now, let's move on to the vector OB. The affix of point B is z_B = √14 - √2i. Following the same procedure as before, we'll use the modulus formula to find the length of OB. In this case, the real part (a) is √14, and the imaginary part (b) is -√2. So, we plug these values into the formula |z| = √(a² + b²): |OB| = √((√14)² + (-√2)²) = √(14 + 2) = √16 = 4. Boom! The modulus of the vector OB is 4. This means the distance from the origin to point B is exactly 4 units. Imagine drawing a line from the origin to the point B on the complex plane. The modulus of OB gives us the precise length of that line. In other words, the magnitude of the vector OB is 4, which tells us how far the point B is located from the origin.

Here’s a friendly recap: The modulus of the vector OB is 4. This signifies that the distance from the origin (0, 0) to point B in the complex plane is exactly 4 units. We're using the modulus formula, and the Pythagorean theorem, to measure the length of the line connecting the origin to point B. It is like measuring the path, or displacement from the start to the end. It's just another way of describing the same geometric concept, but now with a complex twist!

Calculating the Modulus of OC

Okay, almost there! Let's calculate the modulus of the vector OC. The affix of point C is z_C = -1. Notice something? The imaginary part is missing. That's because -1 can be written as -1 + 0i. This means that point C lies on the real axis of the complex plane. The real part (a) is -1, and the imaginary part (b) is 0. Applying the modulus formula, we get: |OC| = √((-1)² + 0²) = √(1 + 0) = √1 = 1. So, the modulus of the vector OC is 1. The distance from the origin to point C is 1 unit. This makes sense, because point C is located at -1 on the real axis. It is literally just one unit away from the origin. The modulus gives us the length of the line segment that connects the origin (0,0) and the point C. Remember, the modulus is always a non-negative real number.

To put it simply: the modulus of the vector OC is 1. This means the distance between the origin and point C is one unit. Because the complex affix is -1, point C is exactly one unit away from the origin along the real axis. This illustrates how the modulus of a complex number can directly translate to the distance from the origin in the complex plane. We have successfully found the modulus of all the vectors!

Conclusion: Wrapping Things Up

Awesome work, guys! We've successfully calculated the moduli of the vectors OA, OB, and OC. Here's a quick recap of our findings:

• |OA| = √7
• |OB| = 4
• |OC| = 1

We started with complex affixes and used the modulus formula, which is rooted in the Pythagorean theorem, to determine the lengths of the vectors. It is a fantastic example of how complex numbers and geometric concepts are intertwined. You've now gained a solid understanding of how to find the modulus of vectors in the complex plane. You're now equipped to tackle similar problems. Keep practicing and exploring – you're doing great!

This exercise highlights the beauty of mathematics, where different concepts come together in a neat package. By understanding complex numbers, affixes, and the modulus, you can solve geometric problems. So, the next time you encounter complex numbers, you will know exactly what to do! Keep up the amazing work! If you have any questions, feel free to ask. Cheers!