Calculating The Magnitude Of Vector Sum A(3, 4) And B(1, 2)

Hey guys! Let's dive into the fascinating world of vectors and explore how to calculate the magnitude of the sum of two vectors. This is a fundamental concept in physics and mathematics, and it's super useful in many real-world applications. Today, we're tackling a specific problem: finding the magnitude of the sum of vectors A(3, 4) and B(1, 2) in the Cartesian plane. It sounds a bit intimidating, but trust me, we'll break it down step by step, making it crystal clear. So, let's put on our mathematical thinking caps and get started!

Understanding Vectors and the Cartesian Plane

Before we jump into calculations, let's quickly recap what vectors are and how they behave in the Cartesian plane. Vectors, at their core, are mathematical entities that possess both magnitude (length) and direction. Think of them as arrows pointing in a specific direction with a certain length. This is in stark contrast to scalars, which only have magnitude (like temperature or speed). Vectors are the backbone of many physical concepts, representing forces, velocities, displacements, and much more.

The Cartesian plane, often called the x-y plane, is our trusty two-dimensional coordinate system. It's formed by two perpendicular lines, the x-axis (horizontal) and the y-axis (vertical), intersecting at a point called the origin (0, 0). Any point in this plane can be uniquely identified by its coordinates (x, y), which tell us how far to move along the x-axis and then along the y-axis to reach that point. This plane provides a visual and mathematical framework for representing and manipulating vectors.

In the context of the Cartesian plane, a vector can be represented by its components. For instance, the vector A(3, 4) tells us that to reach the tip of the vector (starting from the origin), we need to move 3 units along the x-axis and 4 units along the y-axis. Similarly, vector B(1, 2) means we move 1 unit along the x-axis and 2 units along the y-axis. This component representation is crucial for performing mathematical operations on vectors, especially addition, which we'll see in the next section.

Vector Addition: The Key to Finding the Resultant

Now comes the exciting part: adding vectors! When we add vectors, we're essentially finding the resultant vector, which represents the combined effect of the individual vectors. Imagine two forces acting on an object; the resultant vector would represent the net force acting on the object. There are a couple of ways to visualize vector addition, but the most common and intuitive method is the parallelogram law or the head-to-tail method.

The head-to-tail method is particularly straightforward. To add vector B to vector A, you simply place the tail of vector B at the head of vector A. The resultant vector then stretches from the tail of vector A to the head of vector B. It's like following a path: first, you move along vector A, and then you continue along vector B. The resultant vector represents the direct path from your starting point to your final destination.

However, for calculations, we often prefer a more algebraic approach. When vectors are given in component form, vector addition becomes incredibly simple: we just add the corresponding components. That is, if A = (Ax, Ay) and B = (Bx, By), then the resultant vector C = A + B is given by C = (Ax + Bx, Ay + By). This is a powerful technique because it allows us to perform vector addition without having to draw diagrams, making it much more efficient for complex problems.

Calculating the Magnitude: How Long is the Resultant?

Once we've found the resultant vector, the next step is to calculate its magnitude. The magnitude, often denoted by ||C||, represents the length of the vector. In the context of our problem, it tells us the overall effect of combining vectors A and B. Think of it as the total distance covered by the resultant vector.

The beauty of the Cartesian plane is that it provides us with a straightforward way to calculate magnitudes using the Pythagorean theorem. Remember that a vector in the Cartesian plane can be thought of as the hypotenuse of a right-angled triangle, where the components of the vector form the legs of the triangle. Therefore, if C = (Cx, Cy), then the magnitude of C is given by:

||C|| = √(Cx² + Cy²)

This formula is the cornerstone of magnitude calculations. It tells us that to find the magnitude, we simply square each component, add the squares together, and then take the square root of the result. This might seem like a lot of steps, but it's a very systematic and reliable way to determine the length of any vector in the Cartesian plane. So, let's put this into action and solve our problem!

Solving the Problem: Finding the Magnitude of A + B

Alright, let's get back to our original problem: finding the magnitude of the sum of vectors A(3, 4) and B(1, 2). We've laid the groundwork, understanding vectors, the Cartesian plane, vector addition, and magnitude calculations. Now, it's time to put all those pieces together and find the solution.

First, we need to find the resultant vector C, which is the sum of A and B. Using the component-wise addition rule, we have:

C = A + B = (3, 4) + (1, 2) = (3 + 1, 4 + 2) = (4, 6)

So, the resultant vector C has components (4, 6). This means that the combined effect of vectors A and B is a vector that moves 4 units along the x-axis and 6 units along the y-axis. But we're not done yet! We need to find the magnitude of this resultant vector, which will give us the length of this combined effect.

Now, we apply the magnitude formula we discussed earlier:

||C|| = √(Cx² + Cy²) = √(4² + 6²) = √(16 + 36) = √52

Okay, we've got √52. But to match the answer choices, we need to simplify this radical. We can factor out the largest perfect square from 52, which is 4. So, we have:

√52 = √(4 * 13) = √4 * √13 = 2√13

Unfortunately, 2√13 isn't one of our options. It seems we need to approximate the value of √52 to find the closest answer. Since √49 = 7 and √64 = 8, we know that √52 is somewhere between 7 and 8. A quick estimation suggests that √52 is closer to 7 than 8. Using a calculator, we find that √52 is approximately 7.21.

Looking at our options (A) 5, (B) 7, (C) 6, (D) 8, the closest value to 7.21 is 7. Therefore, the correct answer is (B) 7.

Conclusion: Mastering Vector Magnitude

Woohoo! We did it! We successfully calculated the magnitude of the sum of two vectors. We navigated through the concepts of vectors, the Cartesian plane, vector addition, and magnitude calculations. This is a powerful skill that will come in handy in many areas of math and physics. You can apply this knowledge to solve a wide range of problems, from finding the resultant force on an object to determining the displacement of a moving particle.

The key takeaways from this problem are:

1. Vectors have both magnitude and direction, and they can be represented by components in the Cartesian plane.
2. Vector addition is performed component-wise: add the corresponding components of the vectors.
3. The magnitude of a vector is its length and can be calculated using the Pythagorean theorem: ||C|| = √(Cx² + Cy²).

So, the next time you encounter a problem involving vectors, remember these principles, and you'll be well-equipped to tackle it. Keep practicing, keep exploring, and keep those mathematical muscles strong! You guys rock!