Scalar Multiplication: Find K For Vector Transformation

Hey guys! Let's dive into a cool problem involving vectors and scalars. We're going to figure out how to find the scalar that transforms one vector into another. This is a fundamental concept in linear algebra, and once you get the hang of it, you'll be solving these problems like a pro. So, let's jump right in!

The Problem: Unveiling the Scalar Transformation

We're given a vector, v = <12, 8>, and we know that Jessie multiplies this vector by a scalar, k, to get a new vector, kv = <9, 6>. Our mission, should we choose to accept it (and we totally do!), is to find the value of k. The options we have are:

A. 3/4 B. -3 C. <-3, -2> D. <4/3, 4/3>

Before we even start crunching numbers, let's take a moment to understand what's happening here. A scalar is just a regular number that we multiply a vector by. When we multiply a vector by a scalar, we're essentially scaling its magnitude (length) and potentially changing its direction (if the scalar is negative). In this case, we're looking for the scalar that transforms the vector <12, 8> into <9, 6>. Let's break down how to solve this step by step.

Step 1: Grasping Scalar Multiplication

At its core, scalar multiplication is a pretty straightforward operation. When you multiply a vector by a scalar, you're essentially multiplying each component of the vector by that scalar. So, if we have a vector v = <a, b> and a scalar k, then kv = <ka, kb>. This means each component of the original vector is scaled by the same factor, k. This maintains the vector's direction while adjusting its length. If k is greater than 1, the vector gets longer; if k is between 0 and 1, it gets shorter; and if k is negative, the vector's direction is reversed. Understanding this basic principle is crucial for solving problems like the one we have. We need to figure out what single number, when multiplied by both 12 and 8, will give us 9 and 6 respectively. This is where the next step comes in handy.

Step 2: Setting Up the Equations

Now that we understand the principle of scalar multiplication, we can set up equations to solve for k. We know that kv = <9, 6>, and we also know that v = <12, 8>. So, we can write this out as:

k <12, 8> = <9, 6>

This gives us two separate equations, one for each component of the vector:

• 12k = 9
• 8k = 6

These equations are the key to unlocking the value of k. Notice that we have two equations for a single unknown, which gives us a way to check our answer. If the value of k we find satisfies both equations, we know we're on the right track. If the k values from both equations are different, it means there's no single scalar that can transform the original vector into the new one, which isn't the case here, but it's a good thing to keep in mind for other problems. Now, let's move on to solving these equations.

Step 3: Solving for k

Alright, let's solve those equations we set up! We have:

• 12k = 9
• 8k = 6

Let's start with the first equation. To isolate k, we divide both sides of the equation by 12:

k = 9 / 12

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

k = (9 ÷ 3) / (12 ÷ 3) = 3 / 4

So, from the first equation, we get k = 3/4. Now, let's check the second equation to make sure this value of k works. We divide both sides of the second equation by 8:

k = 6 / 8

Again, we can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

k = (6 ÷ 2) / (8 ÷ 2) = 3 / 4

Great! We got the same value for k from both equations. This confirms that our solution is consistent and likely correct. It's always a good idea to double-check like this, especially in math problems. Now that we've confidently found the value of k, let's see which answer choice matches our result.

Step 4: The Final Answer

We've crunched the numbers, solved the equations, and found that k = 3/4. Now, let's look back at our answer choices:

A. 3/4 B. -3 C. <-3, -2> D. <4/3, 4/3>

It's clear that the correct answer is A. 3/4. We've successfully found the scalar that transforms the vector <12, 8> into <9, 6>. Give yourself a pat on the back – you've tackled a scalar multiplication problem like a champ! This type of problem is a great example of how algebraic techniques can be used in vector operations, and mastering it will definitely help you in more advanced topics. Remember, the key is to understand the basic principles, set up the equations carefully, and double-check your work. Now, let's recap the whole process to make sure we've got it all down.

Recapping the Solution: Mastering Scalar Multiplication

Okay, let's quickly recap what we did to solve this problem. This will help solidify your understanding and make you even more confident in tackling similar problems in the future. First, we understood scalar multiplication. We recognized that multiplying a vector by a scalar means multiplying each component of the vector by that scalar. This changes the magnitude (length) of the vector and may change its direction if the scalar is negative. Next, we set up equations. We used the given information to create two equations, one for each component of the vector. These equations represented the relationship between the original vector, the scalar, and the resulting vector. Then, we solved for k. We solved each equation independently and found that both gave us the same value for k, which confirmed our solution. Finally, we identified the correct answer by matching our calculated value of k with the answer choices provided. This step is crucial to ensure you answer the specific question being asked. By following these steps, you can confidently solve scalar multiplication problems and build a strong foundation in linear algebra. Remember, practice makes perfect, so try solving similar problems to reinforce your understanding.

Why Scalar Multiplication Matters: Real-World Applications

You might be thinking, "Okay, this is cool, but why does scalar multiplication matter in the real world?" Great question! Scalar multiplication isn't just some abstract mathematical concept; it has tons of practical applications in various fields. For instance, in computer graphics, scalar multiplication is used to scale objects up or down, making them bigger or smaller on the screen. Think about zooming in on a picture or resizing a window – that's scalar multiplication in action! In physics, scalar multiplication is used to calculate forces and velocities. For example, if you double the force applied to an object (multiplying the force vector by the scalar 2), you'll change its motion. In engineering, scalar multiplication is used in structural analysis to determine how loads affect different parts of a structure. By scaling force vectors, engineers can predict how a bridge or building will respond to varying stresses. Even in economics, scalar multiplication can be used to scale economic indicators, such as multiplying a country's GDP by a growth factor. These are just a few examples, but they show how fundamental scalar multiplication is to many aspects of our lives. Understanding this concept opens up a world of possibilities and allows you to apply math to real-world problems. So, the next time you're resizing an image or seeing a building being constructed, remember the power of scalar multiplication!

Practice Problems: Sharpening Your Skills

Now that we've tackled a scalar multiplication problem together and discussed its real-world applications, it's time to put your skills to the test! Practice makes perfect, and the more problems you solve, the more confident you'll become. Here are a couple of practice problems to get you started:

1. If v = <3, -2> and kv = <12, -8>, what is k?
2. If v = <-1, 5> and k = -2, what is kv?

Try solving these problems on your own, and don't be afraid to revisit the steps we discussed earlier. Remember to first understand the concept, then set up the equations, solve for the unknown, and finally, check your answer. If you get stuck, try breaking down the problem into smaller steps or looking back at the examples we've covered. Working through these practice problems will not only improve your understanding of scalar multiplication but also boost your problem-solving skills in general. Remember, math is like a muscle – the more you exercise it, the stronger it gets! So, keep practicing, keep exploring, and you'll be amazed at what you can achieve. And hey, if you have any questions or want to discuss your solutions, feel free to reach out. We're all in this together, and learning is always more fun when we collaborate!