In the realm of mathematics, particularly in vector algebra, understanding vector operations is paramount. Guys, we're going to dive into the fascinating world of vector subtraction today, breaking down a problem step by step to make sure we've got a solid grasp of how it all works. We'll tackle a detailed example that involves unit vectors, which are super important for representing directions in space. Get ready to boost your math skills and confidently navigate the complexities of vector manipulation!
Let's consider the vectors a, b, c, and d defined in terms of the unit vectors i and j:
The mission, should you choose to accept it, is to compute the expression $(a - b) - (2d - c)$ and express the result in terms of i and j. This problem is a fantastic exercise in vector arithmetic, and by the end of this guide, you'll be a pro at handling similar calculations. We're going to break it down bit by bit, so don't sweat it if it looks intimidating at first!
Step 1: Compute (a - b)
The initial move is to find the difference between vectors a and b. To subtract vectors, we subtract their corresponding components. Remember, vectors are just ordered lists of numbers, and we treat each position separately. So, for a - b, we subtract the i components and the j components individually. This is a fundamental operation, and it's crucial for getting the correct answer. We're setting the stage for a smooth solution by nailing this first step.
To make things crystal clear, let's distribute the negative sign in the second vector. This means flipping the signs of both the i and j components inside the parentheses. This simple step is essential to avoid errors in the subsequent calculations. Trust me, taking it slow and steady here pays off in the long run.
Now, we'll gather the like terms together. We'll group the i terms and the j terms so we can combine them. This is just like simplifying algebraic expressions – we're collecting similar elements to make the expression neater and easier to manage. It's a classic technique that works wonders in vector algebra too.
Performing the additions and subtractions, we get:
So, we've found that a - b equals 6i + 4j. This is a significant milestone in our calculation, and we're one step closer to the final answer. We've handled the first part of the expression, and now we'll move on to the second part with equal precision.
Step 2: Compute (2d - c)
Next, we need to determine the value of (2d - c). This involves a scalar multiplication and another vector subtraction. Scalar multiplication is when we multiply a vector by a regular number (a scalar), and it's a straightforward process. We simply multiply each component of the vector by the scalar. This scales the vector, changing its magnitude but not its direction.
First, let's calculate 2d. Given that d = 14j, we have:
Now, we'll subtract vector c from 2d. Remember, c = i - 3j. We're applying the same principle of vector subtraction as before, subtracting corresponding components. Precision is key, and we're ensuring we get every sign and coefficient right.
Again, distributing the negative sign is crucial:
Grouping the like terms (in this case, the j terms), we have:
Performing the addition, we obtain:
So, 2d - c = -i + 31j. We've successfully computed this part of the expression, and we're on the home stretch now. We're carefully building up our solution, step by step, ensuring we have a clear and correct path to the final answer.
Step 3: Compute (a - b) - (2d - c)
Finally, we're ready to compute the entire expression: (a - b) - (2d - c). We've already calculated both (a - b) and (2d - c), so now it's just a matter of subtracting the second result from the first. This is the final step in our journey, and we're bringing together all our previous work to reach the destination.
We know that (a - b) = 6i + 4j and (2d - c) = -i + 31j. So, we have:
Distribute the negative sign:
Now, let's group the like terms:
Finally, perform the additions and subtractions:
Therefore, (a - b) - (2d - c) = 7i - 27j. We've successfully navigated the vector operations and arrived at our final answer. This problem showcases the importance of methodical calculation and attention to detail in vector algebra. You guys did great!
In this guide, we've meticulously worked through a vector subtraction problem, emphasizing each step for clarity. Understanding these operations is crucial for further studies in physics, engineering, and computer graphics. Vector subtraction, like any mathematical operation, becomes second nature with practice. Keep tackling problems, and you'll become more and more confident in your abilities. The key is to break down complex problems into manageable steps, just like we did here. Remember, every journey starts with a single step, and in math, every problem has a solution waiting to be discovered. Keep exploring, keep learning, and most importantly, keep having fun with math!