Vector Resultant: Solve Z = 2(2,7,-4) - (1,1,1) + 3(2,0,0)
Hey guys! Let's dive into a cool vector problem today. We need to figure out the resultant vector z from the operation z = 2(2,7,-4) - (1,1,1) + 3(2,0,0). This might seem a bit daunting at first, but don't worry, we'll break it down step by step. Vectors are fundamental in various fields like physics, engineering, and computer graphics, so understanding how to manipulate them is super important. This problem involves scalar multiplication and vector addition/subtraction, which are key operations in linear algebra. So, grab your thinking caps, and let's get started!
Understanding Vector Operations
Before we jump into the calculation, let's quickly recap the vector operations involved here. We're dealing with two main types: scalar multiplication and vector addition/subtraction. Scalar multiplication is when you multiply a vector by a scalar (a regular number). This simply scales the magnitude of the vector. For example, if you have a vector (a, b, c) and you multiply it by a scalar k, you get (ka, kb, kc). Vector addition/subtraction is when you add or subtract vectors component-wise. That means you add or subtract the corresponding components of the vectors. For instance, if you have two vectors (a, b, c) and (x, y, z), their sum is (a+x, b+y, c+z), and their difference is (a-x, b-y, c-z).
Scalar Multiplication
Scalar multiplication is a fundamental operation in vector algebra, and it’s pretty straightforward. When you multiply a vector by a scalar, you're essentially scaling the vector. The scalar acts as a multiplier for each component of the vector. So, if you have a vector v = (x, y, z) and a scalar k, the result of kv is (kx, ky, kz). Let’s illustrate this with an example. Suppose we have a vector v = (2, -1, 3) and a scalar k = 2. Multiplying the vector by the scalar, we get 2 * (2, -1, 3) = (22, 2(-1), 2*3) = (4, -2, 6). This means each component of the original vector is multiplied by 2, effectively stretching the vector by a factor of 2 in each dimension. Scalar multiplication doesn't change the direction of the vector if the scalar is positive, but it reverses the direction if the scalar is negative. For instance, if we multiply the same vector v by k = -1, we get -1 * (2, -1, 3) = (-2, 1, -3). This results in a vector pointing in the opposite direction with the same magnitude. Understanding scalar multiplication is crucial because it’s used extensively in various vector operations and transformations. In our original problem, we see scalar multiplication being applied to the vectors (2,7,-4) and (2,0,0), so this concept is directly relevant to finding the solution. Remember, scalars are just numbers, and they help us modify the magnitude (and sometimes the direction) of vectors in a simple and predictable way.
Vector Addition and Subtraction
Vector addition and subtraction are equally essential operations in vector manipulation. These operations are performed component-wise, meaning you add or subtract the corresponding components of the vectors. If you have two vectors, u = (a, b, c) and v = (x, y, z), their sum u + v is calculated by adding the corresponding components: (a+x, b+y, c+z). Similarly, the difference u - v is calculated by subtracting the corresponding components: (a-x, b-y, c-z). Let’s look at an example to clarify this. Suppose u = (1, 2, 3) and v = (4, -5, 6). The sum u + v would be (1+4, 2+(-5), 3+6) = (5, -3, 9). The difference u - v would be (1-4, 2-(-5), 3-6) = (-3, 7, -3). It's important to note that vector addition and subtraction are only defined for vectors of the same dimension. You can’t add or subtract a 2D vector from a 3D vector, for example. Geometrically, vector addition can be visualized using the parallelogram law or the triangle law, where the resultant vector represents the diagonal of the parallelogram formed by the two vectors or the third side of the triangle, respectively. Vector subtraction can be seen as adding the negative of the second vector to the first. In our original problem, we need to perform both vector subtraction and addition after applying scalar multiplication. So, understanding how to correctly add and subtract vectors is critical for arriving at the correct answer. Remember, always perform the operations component-wise to avoid errors, and make sure you’re dealing with vectors of the same dimension.
Solving the Problem Step-by-Step
Now that we've refreshed our understanding of vector operations, let's tackle the problem at hand: z = 2(2,7,-4) - (1,1,1) + 3(2,0,0). First, we'll perform the scalar multiplication. We multiply the vector (2,7,-4) by 2, which gives us (4, 14, -8). Then, we multiply the vector (2,0,0) by 3, resulting in (6, 0, 0). So our equation now looks like this: z = (4, 14, -8) - (1,1,1) + (6, 0, 0). Next, we perform the vector subtraction and addition from left to right. Subtracting (1,1,1) from (4, 14, -8) gives us (4-1, 14-1, -8-1) = (3, 13, -9). Now we add (6, 0, 0) to (3, 13, -9), which gives us (3+6, 13+0, -9+0) = (9, 13, -9). So, the resultant vector z is (9, 13, -9). This matches option D, which is (9,13,-9). We've successfully navigated through scalar multiplication and vector addition/subtraction to find our solution! Remember, breaking down the problem into smaller, manageable steps makes it much easier to solve. And always double-check your calculations to ensure accuracy.
Step 1: Scalar Multiplication
Let's break down the first part of our problem, which involves scalar multiplication. Remember, scalar multiplication means multiplying a vector by a scalar (a regular number). This operation scales the magnitude of the vector without changing its direction (unless the scalar is negative, in which case the direction is reversed). In our equation, z = 2(2,7,-4) - (1,1,1) + 3(2,0,0), we have two scalar multiplications to perform: 2(2,7,-4) and 3(2,0,0). First, let’s calculate 2(2,7,-4). We multiply each component of the vector (2,7,-4) by the scalar 2: 2 * 2 = 4, 2 * 7 = 14, and 2 * -4 = -8. So, 2(2,7,-4) = (4, 14, -8). Next, we calculate 3(2,0,0). We multiply each component of the vector (2,0,0) by the scalar 3: 3 * 2 = 6, 3 * 0 = 0, and 3 * 0 = 0. So, 3(2,0,0) = (6, 0, 0). Now that we've performed the scalar multiplications, our equation looks like this: z = (4, 14, -8) - (1,1,1) + (6, 0, 0). This simplifies our problem by eliminating the scalar multipliers and setting us up for the next step, which is vector addition and subtraction. Mastering scalar multiplication is key because it’s a building block for more complex vector operations. Always remember to multiply each component of the vector by the scalar to get the correct result.
Step 2: Vector Subtraction and Addition
Now that we've completed the scalar multiplication, the next step is to perform the vector subtraction and addition. Our equation is now z = (4, 14, -8) - (1,1,1) + (6, 0, 0). We’ll perform these operations from left to right. First, let’s subtract the vector (1,1,1) from (4, 14, -8). To do this, we subtract the corresponding components: 4 - 1 = 3, 14 - 1 = 13, and -8 - 1 = -9. So, (4, 14, -8) - (1,1,1) = (3, 13, -9). Now our equation looks like this: z = (3, 13, -9) + (6, 0, 0). Next, we add the vector (6, 0, 0) to (3, 13, -9). Again, we add the corresponding components: 3 + 6 = 9, 13 + 0 = 13, and -9 + 0 = -9. So, (3, 13, -9) + (6, 0, 0) = (9, 13, -9). Therefore, the resultant vector z is (9, 13, -9). This completes the calculation, and we've found our solution by performing scalar multiplication followed by vector subtraction and addition. Remember, the order of operations matters, and performing these operations component-wise ensures we arrive at the correct answer. Vector subtraction and addition are fundamental in various applications, including physics (e.g., calculating resultant forces) and computer graphics (e.g., transforming objects in 3D space).
The Final Result
Alright, we've gone through the entire process, and it's time to state the final result. After performing the scalar multiplication and then the vector subtraction and addition, we found that the resultant vector z is (9, 13, -9). This corresponds to option D in the given choices. So, the correct answer is D. (9,13,-9). Yay, we did it! Understanding how to manipulate vectors is super useful, especially if you're into fields like physics, engineering, or computer graphics. Vectors help us represent quantities that have both magnitude and direction, and mastering vector operations allows us to solve a wide range of problems. Whether you're calculating the trajectory of a projectile, determining the net force on an object, or transforming objects in a 3D game, vectors are your friends. Remember, the key to solving these problems is to break them down into smaller, manageable steps. Start with scalar multiplication, then perform vector addition and subtraction component-wise. And always double-check your work to make sure you haven't made any calculation errors. With practice, you'll become a vector pro in no time!
Conclusion
So, to wrap things up, we successfully determined the resultant vector z from the operation z = 2(2,7,-4) - (1,1,1) + 3(2,0,0). By methodically applying scalar multiplication and vector addition/subtraction, we arrived at the solution z = (9, 13, -9), which is option D. Hopefully, this step-by-step explanation has helped you understand the process involved in solving such problems. Remember, practice makes perfect, so keep working on similar problems to solidify your understanding of vector operations. Vectors are powerful tools, and the more comfortable you are with them, the better equipped you'll be to tackle a variety of challenges in math, science, and engineering. If you found this helpful, keep an eye out for more problem-solving sessions, and feel free to ask any questions you might have. Keep up the great work, guys!
