Finding The Unit Vector For Vector A = 5i - 12j A Step-by-Step Guide
Hey guys! Let's dive into the exciting world of vectors, specifically focusing on how to find the unit vector in the direction of a given vector. Today, we're tackling a classic problem: finding the unit vector for vector A = 5î - 12ĵ. This is a fundamental concept in physics and mathematics, so understanding it is super important for anyone studying these fields. Whether you're a student just starting out or someone looking to brush up on your skills, this guide will walk you through the process step by step. We'll break down the concepts, explain the formulas, and show you exactly how to solve this problem. So, grab your pencils, and let's get started!
Understanding Vectors and Unit Vectors
Before we jump into the solution, let's make sure we're all on the same page about what vectors and unit vectors actually are. This foundational knowledge will make the rest of the process much clearer. So, what exactly is a vector? A vector is a quantity that has both magnitude (or size) and direction. Think of it as an arrow pointing in a certain direction; the length of the arrow represents the magnitude, and the direction the arrow points is, well, the direction! Vectors are used all the time in physics and engineering to represent things like force, velocity, and displacement. For instance, if you're pushing a box across the floor, the force you're applying can be represented as a vector with a magnitude (how hard you're pushing) and a direction (the way you're pushing). Vectors are typically represented in component form. In two dimensions, like in our problem today, we use the unit vectors î and ĵ. The unit vector î points along the positive x-axis, and ĵ points along the positive y-axis. So, a vector like A = 5î - 12ĵ means we have a component of 5 units in the x-direction and -12 units in the y-direction. Now, let's talk about unit vectors. A unit vector is a special kind of vector that has a magnitude of exactly 1. Its primary purpose is to show direction. You can think of it as a normalized version of a vector. Normalizing a vector means scaling it down (or up) until its magnitude is 1, while keeping its direction the same. Unit vectors are super handy because they allow us to specify a direction without worrying about magnitude. We often use unit vectors to represent directions in space, making calculations and problem-solving much easier. In our problem, we want to find the unit vector in the direction of A = 5î - 12ĵ. This means we need to find a vector that points in the same direction as A but has a magnitude of 1. Understanding this distinction between vectors and unit vectors is crucial for solving this type of problem. So, with this knowledge in hand, let's move on to the next step: calculating the magnitude of vector A. This is a key part of finding the unit vector, so pay close attention!
Calculating the Magnitude of Vector A
Alright, now that we know what vectors and unit vectors are, let's get down to the nitty-gritty of calculating the magnitude of our vector A = 5î - 12ĵ. The magnitude of a vector is essentially its length. It tells us how "big" the vector is, regardless of its direction. Think of it as the straight-line distance from the starting point to the ending point of the vector. In mathematical terms, the magnitude of a vector is a scalar quantity (a number) and is always non-negative. To find the magnitude of a vector in two dimensions, we use the Pythagorean theorem. Remember that old friend from geometry class? It turns out it's super useful in vector calculations too! If we have a vector A = xî + yĵ, where x and y are the components of the vector along the x and y axes, respectively, then the magnitude of A, often written as |A| or ||A||, is given by the formula: |A| = √(x² + y²). This formula comes directly from the Pythagorean theorem. Imagine a right triangle where the x and y components of the vector are the lengths of the two legs, and the vector itself is the hypotenuse. The magnitude of the vector is then the length of the hypotenuse, which we find using the theorem. Now, let's apply this formula to our vector A = 5î - 12ĵ. Here, x = 5 and y = -12. So, to find the magnitude |A|, we plug these values into the formula: |A| = √((5)² + (-12)²) = √(25 + 144) = √169. And what's the square root of 169? It's 13! So, the magnitude of vector A is |A| = 13. This means that the length of the vector A is 13 units. This calculation is a crucial step in finding the unit vector because we'll use this magnitude to normalize the vector. Remember, a unit vector has a magnitude of 1, so we need to scale our vector A down until it has this magnitude. Now that we've found the magnitude of A, we're one step closer to finding its unit vector. Next up, we'll use this magnitude to actually calculate the unit vector. So, stick around and let's keep going!
Determining the Unit Vector
Okay, we've calculated the magnitude of vector A, which is 13. Now comes the exciting part: actually determining the unit vector! Remember, the unit vector is a vector with a magnitude of 1 that points in the same direction as our original vector. So, how do we find this magical vector? The process is actually quite straightforward. To find the unit vector in the direction of a vector, we simply divide the vector by its magnitude. Mathematically, if we have a vector A, the unit vector in the direction of A, often denoted as  (pronounced "A-hat"), is given by the formula:  = A / |A|. This formula is the key to solving our problem. It tells us that to get the unit vector, we need to take each component of the original vector and divide it by the magnitude we calculated earlier. Let's apply this to our vector A = 5î - 12ĵ, which has a magnitude of |A| = 13. To find the unit vector Â, we'll divide each component of A by 13:  = (5î - 12ĵ) / 13. We can rewrite this as:  = (5/13)î - (12/13)ĵ. And there you have it! This is the unit vector in the direction of vector A. Notice that we've essentially scaled down the original vector so that its magnitude is now 1, but it still points in the same direction. To double-check our work, we can calculate the magnitude of Â. If we did everything correctly, it should be 1: |Â| = √((5/13)² + (-12/13)²) = √(25/169 + 144/169) = √(169/169) = √1 = 1. Yep, it checks out! The magnitude of our unit vector is indeed 1. This confirms that we've successfully found the unit vector in the direction of A. This process of finding a unit vector is called normalization, and it's a fundamental operation in vector algebra. Unit vectors are super useful because they allow us to represent directions without worrying about magnitude. They're used extensively in physics, engineering, computer graphics, and many other fields. Now that we've found the unit vector for A, let's move on to identifying the correct answer from the given options.
Identifying the Correct Answer
Alright, we've done the hard work and calculated the unit vector in the direction of A = 5î - 12ĵ. We found that the unit vector  is (5/13)î - (12/13)ĵ. Now, it's time to match our result with the given options and identify the correct answer. This is a crucial step to make sure we've solved the problem completely and haven't made any silly mistakes along the way. Let's take a look at the options provided:
(A) î (B) ĵ (C) (î + ĵ) / 13 (D) (5î - 12ĵ) / 13
Now, let's compare our calculated unit vector,  = (5/13)î - (12/13)ĵ, with these options. Option (A) is simply î, which is the unit vector in the x-direction. This is clearly not the same as our result, so we can eliminate this option. Option (B) is ĵ, the unit vector in the y-direction. Again, this doesn't match our unit vector, so we can rule this out as well. Option (C) is (î + ĵ) / 13. This vector has components in both the x and y directions, but the components are not the same as in our calculated unit vector. So, this option is also incorrect. Finally, option (D) is (5î - 12ĵ) / 13. This looks very familiar! In fact, it's exactly the same as our calculated unit vector  = (5/13)î - (12/13)ĵ. The only difference is that it's written in a slightly different form, but it represents the same vector. So, we can confidently say that option (D) is the correct answer. We've successfully matched our calculated result with the provided options and identified the correct one. This step is important because it ensures that we've not only done the math correctly but also understood what the question was asking and how to express the answer in the required format. So, congratulations! We've solved the problem and found the unit vector in the direction of A. But before we wrap up, let's quickly summarize the steps we took and reinforce the key concepts.
Conclusion: Key Takeaways for Unit Vector Calculations
Awesome! We've successfully navigated the process of finding the unit vector in the direction of vector A = 5î - 12ĵ. Before we wrap things up, let's recap the key takeaways from this exercise. This will help solidify your understanding and make sure you're ready to tackle similar problems in the future. First, remember what a vector and a unit vector are. A vector has both magnitude and direction, while a unit vector is a vector with a magnitude of 1 that points in a specific direction. Unit vectors are super useful for representing directions in space. The core of solving this problem was understanding the formula for finding a unit vector: Â = A / |A|. This formula tells us that to find the unit vector in the direction of a vector A, we simply divide the vector by its magnitude. The steps we followed were:
- Calculate the magnitude of the vector A using the Pythagorean theorem: |A| = √(x² + y²).
- Divide the vector A by its magnitude |A| to find the unit vector Â.
- Double-check your work by calculating the magnitude of the unit vector Â. It should be equal to 1.
- Match your calculated unit vector with the given options to identify the correct answer.
By following these steps, you can confidently find the unit vector for any given vector. This is a fundamental skill in physics and mathematics, and mastering it will open doors to solving a wide range of problems. Remember, practice makes perfect! The more you work with vectors and unit vectors, the more comfortable you'll become with the concepts and the calculations. So, don't be afraid to tackle more problems and challenge yourself. With a solid understanding of vectors and unit vectors, you'll be well-equipped to excel in your studies and in any field that requires these concepts. Keep up the great work, and happy calculating!
