Vector Sum: A = (3, 4) And B = (1, 2) - Correct Answer

Hey everyone! Let's dive into the world of vector operations and specifically focus on how to correctly calculate the sum of two vectors. This is a fundamental concept in mathematics and physics, and understanding it well is crucial for tackling more complex problems. In this article, we'll break down the process step by step, ensuring you grasp the concept with ease. We'll use the example of finding the sum of vectors A = (3, 4) and B = (1, 2) to illustrate the process. So, grab your thinking caps, and let's get started!

Understanding Vectors and Vector Addition

Before we jump into the calculation, let's quickly recap what vectors are and how vector addition works. Vectors, unlike simple numbers (scalars), have both magnitude (length) and direction. Think of them as arrows pointing in a specific way. They are often represented as ordered pairs (in two dimensions) or triples (in three dimensions), like our A = (3, 4) and B = (1, 2). These numbers represent the vector's components along the x and y axes, respectively.

Now, when we add vectors, we're essentially combining their effects. Imagine two forces pulling an object – the resulting force is the vector sum of the individual forces. Mathematically, vector addition is quite straightforward: you simply add the corresponding components of the vectors. This means you add the x-components together and the y-components together (and the z-components if you're dealing with 3D vectors). This component-wise addition ensures that we're accounting for both the magnitude and direction of each vector.

To make it clearer, let's express this in a general formula. If we have two vectors, let's say u = (u₁, u₂) and v = (v₁, v₂), their sum, u + v, would be calculated as:

u + v = (u₁ + v₁, u₂ + v₂)

This formula might seem simple, but it's the foundation of vector addition. It highlights the importance of adding corresponding components, which is key to getting the correct resultant vector. It's a fundamental operation in fields like physics and engineering where vectors are used to represent forces, velocities, and other quantities that have both magnitude and direction. Understanding this concept well will pave the way for tackling more complex vector operations, such as scalar multiplication and dot products, so make sure to grasp this thoroughly.

Step-by-Step Calculation of A + B

Now that we've laid the groundwork, let's apply the concept to our specific example: finding the sum of vectors A = (3, 4) and B = (1, 2). We'll break down the calculation into clear, easy-to-follow steps. This will not only give you the answer but also reinforce your understanding of the process.

Step 1: Identify the components of each vector.

We've already stated them, but let's reiterate for clarity: A = (3, 4) and B = (1, 2). This means vector A has an x-component of 3 and a y-component of 4. Vector B has an x-component of 1 and a y-component of 2. Identifying these components correctly is the first crucial step in vector addition. It's like labeling the ingredients before you start cooking – you need to know what you're working with!

Step 2: Add the corresponding components.

This is where the vector addition magic happens! As we discussed earlier, we add the x-components together and the y-components together. So, we have:

• x-component of A + x-component of B = 3 + 1 = 4
• y-component of A + y-component of B = 4 + 2 = 6

See how we paired the components? The x-component of A was added to the x-component of B, and similarly for the y-components. This is essential for maintaining the directional information encoded in the vectors. A common mistake is to mix and match components, which would lead to an incorrect result. Remember, it's always corresponding components that get added together.

Step 3: Write the resulting vector.

Now that we've added the components, we can express the sum of the vectors, A + B, as a new vector. Using the results from Step 2, we get:

A + B = (4, 6)

And there you have it! The sum of vectors A and B is the vector (4, 6). This new vector represents the combined effect of A and B. If A and B were forces, (4, 6) would represent the resultant force. If they were velocities, (4, 6) would represent the combined velocity. This final step is about putting the pieces together and expressing the result in the correct vector notation.

Analyzing the Options

Now that we've calculated A + B, let's circle back to the original question and analyze the given options. We were asked to identify the option that correctly represents the sum of vectors A = (3, 4) and B = (1, 2). We've already determined that A + B = (4, 6).

Let's take a look at the options:

a) A + B = (4, 6) b) A + B = (2, 3) c) A + B = (3, 2) d) A + B = (1, 4)

By comparing our calculated result (4, 6) with the options, it's clear that option a) A + B = (4, 6) is the correct one. The other options present incorrect sums, likely resulting from adding the components incorrectly or mixing them up. This highlights the importance of following the component-wise addition rule carefully.

Choosing the correct option isn't just about getting the right answer; it's about demonstrating your understanding of the underlying concept. In this case, it shows you understand how vector addition works and how to apply it correctly. So, when you're faced with similar questions, always take the time to do the calculation step by step, and then confidently select the option that matches your result.

Why is Vector Addition Important?

You might be wondering,