Vector Operation: Solve For Resultant Vector Z

by Ahmed Latif 47 views

Hey everyone! Today, we're diving into the fascinating world of vector operations. We've got a problem that looks a bit like a mathematical puzzle, but don't worry, we'll break it down step by step. Our goal is to determine the resultant vector z from the operation: z = 2 ⋅ (2, 7, -4) - (1, 1, 1) + 3 ⋅ (2, 0, 0). Stick with me, and we'll solve this together!

Understanding Vector Operations

Before we jump into the solution, let's quickly recap what vector operations are all about. Vectors, unlike simple numbers, have both magnitude and direction. Think of them as arrows pointing in space. We can perform several operations on vectors, including scalar multiplication, vector addition, and vector subtraction. Scalar multiplication involves multiplying a vector by a scalar (a regular number), which changes the vector's magnitude but not its direction. Vector addition and subtraction involve combining vectors component-wise. This means we add or subtract the corresponding components of the vectors.

Now, let's break down the given equation, z = 2 ⋅ (2, 7, -4) - (1, 1, 1) + 3 ⋅ (2, 0, 0), into manageable parts. We'll start with the scalar multiplications and then move on to the vector addition and subtraction. This step-by-step approach will make the entire process clear and easy to follow. Remember, the key is to take it one operation at a time, ensuring accuracy at each stage.

Step 1: Scalar Multiplication

Our first task is to handle the scalar multiplications. We have two such operations in our equation: 2 ⋅ (2, 7, -4) and 3 ⋅ (2, 0, 0). To perform scalar multiplication, we simply multiply each component of the vector by the scalar. Let's start with 2 ⋅ (2, 7, -4). Multiplying each component by 2, we get (22, 27, 2(-4)) = (4, 14, -8)*. So, the first scalar multiplication gives us the vector (4, 14, -8). Now, let's move on to the second scalar multiplication: 3 ⋅ (2, 0, 0). Multiplying each component by 3, we get (32, 30, 3*0) = (6, 0, 0). Thus, the second scalar multiplication results in the vector (6, 0, 0). We've successfully completed the scalar multiplications, and our equation now looks like this: z = (4, 14, -8) - (1, 1, 1) + (6, 0, 0). We're one step closer to finding the resultant vector z!

Step 2: Vector Subtraction

Next up, we'll tackle the vector subtraction. Our equation now reads: z = (4, 14, -8) - (1, 1, 1) + (6, 0, 0). We need to subtract the vector (1, 1, 1) from (4, 14, -8). Vector subtraction is performed component-wise, meaning we subtract the corresponding components of the vectors. So, we subtract the first components (4 - 1), the second components (14 - 1), and the third components (-8 - 1). This gives us (4-1, 14-1, -8-1) = (3, 13, -9). Therefore, the result of the vector subtraction is the vector (3, 13, -9). Our equation is now simplified to: z = (3, 13, -9) + (6, 0, 0). We're almost there! We just have one more vector operation to perform.

Step 3: Vector Addition

Finally, we arrive at the last step: vector addition. Our equation is now z = (3, 13, -9) + (6, 0, 0). Just like vector subtraction, vector addition is also performed component-wise. We add the corresponding components of the vectors. So, we add the first components (3 + 6), the second components (13 + 0), and the third components (-9 + 0). This gives us (3+6, 13+0, -9+0) = (9, 13, -9). Thus, the result of the vector addition is the vector (9, 13, -9). And that's it! We've successfully performed all the vector operations.

The Resultant Vector

After meticulously performing the scalar multiplications, vector subtraction, and vector addition, we've arrived at our final answer. The resultant vector z is (9, 13, -9). So, the correct answer is D. (9, 13, -9). Great job to everyone who followed along! Vector operations might seem a bit daunting at first, but by breaking them down into smaller, manageable steps, we can solve even the most complex problems. Remember, practice makes perfect, so keep working on these types of problems to build your skills and confidence.

Analyzing the Options

Let's take a quick look at the other options to understand why they are incorrect. Option A. (-9, -13, -9) has the wrong signs for the first two components. Option B. (9, 0, 0) is incorrect because it doesn't account for the contributions from the 2 ⋅ (2, 7, -4) and (1, 1, 1) terms. Option C. (9, 13, 0) has the correct first two components but an incorrect third component. Option E. (9, 11, 9) has an incorrect second component and an incorrect sign for the third component. By analyzing these options, we can see how important it is to perform each operation carefully and accurately.

Mastering Vector Operations: Tips and Tricks

To truly master vector operations, here are a few tips and tricks that can help you along the way:

  1. Break it Down: As we did in this problem, break down complex equations into smaller, manageable steps. This will help you avoid errors and keep your work organized.
  2. Component-Wise Operations: Remember that vector addition and subtraction are performed component-wise. This means you add or subtract the corresponding components of the vectors.
  3. Scalar Multiplication First: Always perform scalar multiplications before vector addition or subtraction. This follows the order of operations.
  4. Double-Check Your Work: It's always a good idea to double-check your calculations, especially when dealing with negative numbers.
  5. Practice, Practice, Practice: The more you practice vector operations, the more comfortable and confident you'll become.

By following these tips and tricks, you'll be well on your way to mastering vector operations. Remember, the key is to be patient, persistent, and to take it one step at a time.

Why Vector Operations Matter

You might be wondering, why are vector operations so important? Well, vectors and vector operations are fundamental concepts in many areas of science, engineering, and computer graphics. They are used to represent forces, velocities, displacements, and many other physical quantities. In physics, vector operations are used to calculate the resultant force acting on an object, the velocity of a moving object, and the displacement of an object. In computer graphics, vectors are used to represent the vertices of 3D models, the direction of light sources, and the position of the camera. Understanding vector operations is crucial for anyone working in these fields.

For instance, in game development, vectors are used to control the movement of characters, the trajectory of projectiles, and the interaction of objects in the game world. In robotics, vectors are used to control the motion of robots, plan their paths, and avoid obstacles. In machine learning, vectors are used to represent data points, and vector operations are used to perform calculations such as finding the distance between two data points or the average of a set of data points. The applications of vector operations are vast and varied, making it an essential topic for anyone pursuing a career in a STEM field.

Conclusion

So, there you have it! We've successfully navigated the world of vector operations and found the resultant vector z to be (9, 13, -9). We started by understanding the basic concepts of vector operations, then broke down the problem into manageable steps, and finally, we arrived at the solution. Remember, the key to mastering vector operations is to practice, be patient, and take it one step at a time. And remember that guy's vectors are all around us, underpinning countless technologies and scientific principles. Keep exploring, keep learning, and you'll be amazed at what you can achieve!