Finding 'm' For Perpendicular Vectors: A Physics Problem

Hey guys! Let's dive into a cool physics problem today that involves finding the value of 'm' when two vectors are perpendicular. This is a classic question that pops up in vector algebra and is super important for understanding how vectors interact in space. We'll break down the problem step by step, so you can totally nail it. Grab your thinking caps, and let's get started!

Understanding Perpendicular Vectors

Before we jump into the nitty-gritty of the problem, let's quickly recap what it means for vectors to be perpendicular. Perpendicular vectors, also known as orthogonal vectors, are vectors that meet at a right angle (90 degrees). This geometric relationship has a very neat algebraic consequence: the dot product (also known as the scalar product) of two perpendicular vectors is always zero. Remember this, guys; it's the key to solving this problem!

So, mathematically, if we have two vectors, let's say A and B, they are perpendicular if and only if their dot product, A · B, equals zero. The dot product is calculated by multiplying the corresponding components of the vectors and then adding the results. This concept is super crucial for many areas of physics, from mechanics to electromagnetism, so make sure you've got a solid grasp of it. In this problem, we'll use this property to find the value of 'm' that makes the given vectors perpendicular. Mastering this concept opens doors to solving a wide range of physics problems, so let's dig in and get comfortable with it!

Setting Up the Problem

Okay, now that we've got the basics down, let's get to the problem at hand. We're given two vectors: one is i + 2j - 3k, and the other is 3i + mj - 9k. Our mission, should we choose to accept it (and we do!), is to find the value of 'm' that makes these two vectors perpendicular. To do this, we're going to use that handy fact we just talked about: the dot product of perpendicular vectors is zero.

First, let's write down the vectors in a more formal notation. We can represent the first vector, A, as (1, 2, -3) and the second vector, B, as (3, m, -9). These are the components of the vectors along the x, y, and z axes, respectively. Writing them down this way makes it super clear when we calculate the dot product. Next, we'll set up the dot product equation, which is A · B = 0. This equation is our roadmap for solving the problem. We'll plug in the components of our vectors, do the math, and then solve for 'm'. Setting up the problem correctly is half the battle, guys, so let's make sure we've got this solid before we move on to the calculations.

Calculating the Dot Product

Alright, let's get our hands dirty with some calculations! We know that the dot product of two vectors A (1, 2, -3) and B (3, m, -9) is given by multiplying their corresponding components and then adding the results. So, A · B = (1 * 3) + (2 * m) + (-3 * -9). Let's break this down step by step to make sure we don't miss anything. First, we multiply the i components: 1 * 3 = 3. Then, we multiply the j components: 2 * m = 2m. And finally, we multiply the k components: -3 * -9 = 27. Now, we add these results together: 3 + 2m + 27.

So, our dot product A · B simplifies to 2m + 30. Remember, we know that for the vectors to be perpendicular, this dot product must equal zero. So, we set up the equation 2m + 30 = 0. Now we have a simple algebraic equation to solve for 'm'. We're almost there, guys! A little bit of algebra, and we'll have our answer. Calculating the dot product carefully is super important; it's the heart of the solution. Double-check your work to make sure everything is spot on before moving to the next step. Accuracy here will save you headaches later!

Solving for 'm'

Okay, we've got our equation: 2m + 30 = 0. Now it's time to unleash our algebraic superpowers and solve for 'm'. This is pretty straightforward, guys. First, we want to isolate the term with 'm' in it. So, we subtract 30 from both sides of the equation. This gives us 2m = -30. Next, to get 'm' by itself, we divide both sides of the equation by 2. This gives us m = -30 / 2, which simplifies to m = -15.

And there we have it! The value of 'm' that makes the two vectors perpendicular is -15. That wasn't so bad, right? We took the dot product, set it equal to zero, and solved the resulting equation. This is a classic technique in vector algebra, and you'll see it pop up again and again in physics problems. Always remember to take it one step at a time, and don't rush the process. Accuracy is key, especially when dealing with negative signs and fractions. Now, let's take a moment to recap what we've done and make sure we've got a solid understanding of the whole process.

Checking the Answer

Before we pat ourselves on the back and call it a day, let's just quickly check our answer to make sure everything adds up. We found that m = -15. So, our vectors are A = (1, 2, -3) and B = (3, -15, -9). To check if these vectors are indeed perpendicular, we need to calculate their dot product again and see if it equals zero.

Let's do it: A · B = (1 * 3) + (2 * -15) + (-3 * -9) = 3 - 30 + 27. If we add those up, we get 3 - 30 + 27 = 0. Awesome! The dot product is zero, which confirms that the vectors are perpendicular when m = -15. This step is super important, guys. Checking your answer not only ensures you've got the correct solution but also reinforces your understanding of the concepts. It's like a little victory lap for your brain! So, always take the time to check your work; it's a habit that will serve you well in physics and beyond.

Conclusion

So, to wrap things up, we've successfully found the value of 'm' that makes the vectors i + 2j - 3k and 3i + mj - 9k perpendicular. We did this by using the fact that the dot product of perpendicular vectors is zero. We calculated the dot product, set it equal to zero, solved for 'm', and even checked our answer to make sure it was correct. Nice job, guys!

This kind of problem is a fantastic example of how vector algebra can be used to solve real-world physics questions. Understanding perpendicular vectors and the dot product is crucial for tackling more complex topics in physics, such as work, energy, and electromagnetism. The key takeaway here is the connection between the geometric concept of perpendicularity and the algebraic tool of the dot product. Keep practicing these kinds of problems, and you'll become a vector ninja in no time! Remember, physics is all about understanding the fundamental principles and applying them creatively to solve problems. Keep up the great work, and you'll ace those physics challenges!