Finding Vector V Parallel To U With Dot Product -42

Hey everyone! Today, let's dive into a fascinating problem involving vectors. We're tasked with finding a vector v that's parallel to another vector u, given their dot product. This is a classic vector algebra problem that combines the concepts of parallel vectors and dot products. We're going to break it down step by step, so you'll not only understand the solution but also the underlying principles.

Problem Statement

Here’s the problem we’re tackling: Given vector v = (x, y, z) is parallel to vector u = (2, -1, 3), and their dot product v . u = -42, we need to find vector v. This problem is a blend of vector algebra concepts, making it essential to understand both the properties of parallel vectors and the definition of the dot product. So, let's get started and dissect this problem to its core.

Understanding Parallel Vectors

First off, let's talk about what it means for two vectors to be parallel. Parallel vectors are vectors that point in the same direction or opposite directions. The key here is that one vector can be obtained by scaling the other. Mathematically, if v is parallel to u, then v = k * u, where k is a scalar. This scalar, k, is super important because it's the link between the two parallel vectors. It tells us how much u needs to be stretched or shrunk (and potentially flipped) to become v. So, in our case, since v is parallel to u = (2, -1, 3), we can express v as (2k, -k, 3k). This gives us a solid foundation to start solving our problem. The concept of parallel vectors is not just a mathematical abstraction; it has real-world applications in physics, engineering, and computer graphics, where understanding directional relationships is crucial.

Now, let's see how the dot product comes into play.

Dot Product and Its Role

The dot product is a fundamental operation in vector algebra. For two vectors, a = (a1, a2, a3) and b = (b1, b2, b3), the dot product a . b is calculated as a1b1 + a2b2 + a3*b3. In simpler terms, you multiply the corresponding components and add them up. But what does this number actually tell us? The dot product has a geometric interpretation: a . b = |a| |b| cos(θ), where |a| and |b| are the magnitudes (lengths) of the vectors, and θ is the angle between them. This formula reveals a lot. For instance, if the dot product is zero, it means cos(θ) is zero, implying the vectors are perpendicular (θ = 90 degrees). When the dot product is positive, the angle is acute (less than 90 degrees), and when it's negative, the angle is obtuse (greater than 90 degrees). In our problem, we're given that v . u = -42. This negative value tells us that the angle between v and u is obtuse, meaning they point in generally opposite directions. The dot product is a powerful tool, providing insights into the relationship between vectors, and it's crucial for solving problems like ours.

Let's use this knowledge to find the value of k.

Solving for the Scalar Multiple k

Okay, we know that v = (2k, -k, 3k) and u = (2, -1, 3), and we're given that v . u = -42. Now we can actually calculate the dot product using the components and set it equal to -42. This gives us:

(2k * 2) + (-k * -1) + (3k * 3) = -42

Let's simplify this equation:

4k + k + 9k = -42

Combining the terms, we get:

14k = -42

Now, we can solve for k by dividing both sides by 14:

k = -42 / 14 k = -3

So, the scalar multiple k is -3. This means vector v is -3 times vector u. This negative sign is important because it confirms that v and u point in opposite directions, which aligns with the negative dot product we were given. Finding k is a critical step, as it allows us to determine the exact components of vector v. With k in hand, we're just one step away from the final answer.

Now, let's find vector v.

Determining Vector v

We've found that k = -3, and we know that v = (2k, -k, 3k). Now it's straightforward to find the components of v by substituting k:

v = (2 * -3, -(-3), 3 * -3)

This simplifies to:

v = (-6, 3, -9)

So, vector v is (-6, 3, -9). This vector is parallel to u and satisfies the dot product condition v . u = -42. We've successfully found the vector that meets all the given criteria. This final step demonstrates how all the pieces of the puzzle fit together: the concept of parallel vectors, the dot product, and the scalar multiple. Each step built upon the previous one, leading us to the solution. Now that we have our answer, let’s recap the whole process.

Conclusion

Alright, guys, we've successfully found vector v! To recap, we started with the fact that v is parallel to u, which allowed us to express v as a scalar multiple of u. Then, we used the dot product information to set up an equation and solve for the scalar k. Finally, we plugged k back into our expression for v to find the vector v = (-6, 3, -9). This problem is a great example of how vector algebra combines different concepts to solve geometric problems. Understanding these concepts is crucial for anyone working with vectors in mathematics, physics, or computer science. Vector problems like these might seem daunting at first, but by breaking them down into smaller steps and understanding the underlying principles, you can tackle even the trickiest ones. Keep practicing, and you'll become a vector whiz in no time!