Hey guys! Today, we're diving into a fun vector problem. We've got three vectors, u, v, and w, and our mission is to find a new vector, z, that's a linear combination of these three. Don't worry, it sounds more complicated than it is. We'll break it down step by step, so you'll be a vector whiz in no time! This comprehensive guide will walk you through the process of finding the vector z when given vectors u, v, and w, and the equation z = 7u - 3v - (1/2)w. We'll cover vector operations, scalar multiplication, and vector subtraction, ensuring you grasp the concepts thoroughly. Let's get started and make math fun together!
Understanding the Problem
Before we jump into the calculations, let's make sure we understand what the problem is asking. We're given three vectors:
- u = <1, 2, 3>
- v = <2, 2, -1>
- w = <4, 0, -4>
And we need to find the vector z, which is defined as:
z = 7u - 3v - (1/2)w
This equation tells us that z is a linear combination of u, v, and w. This means we're going to multiply each vector by a scalar (a number) and then add or subtract the resulting vectors. Think of it like mixing ingredients in a recipe – each vector contributes to the final vector z in a specific proportion. Understanding the problem setup is the first crucial step in solving it accurately. The beauty of vector algebra lies in its ability to represent complex spatial relationships in a concise and manageable form. Here, we're not just dealing with abstract mathematical entities; vectors can represent physical quantities like forces, velocities, and displacements. By manipulating them using scalar multiplication and vector addition/subtraction, we can solve a wide range of problems in physics, engineering, and computer graphics. For example, in computer graphics, vectors are used to define the vertices of 3D models, and transformations like scaling, rotation, and translation are performed using vector operations. In physics, vectors are used to represent forces acting on an object, and the net force can be calculated by adding the individual force vectors. This problem, while seemingly abstract, lays the foundation for understanding these real-world applications. So, as we proceed with the calculations, keep in mind that we're not just crunching numbers; we're learning a powerful tool that can be used to model and solve problems in various fields. The key is to break down the problem into smaller, manageable steps. First, we'll perform the scalar multiplication, then we'll handle the vector addition and subtraction. By taking it one step at a time, we can avoid errors and ensure that we arrive at the correct solution. Remember, math is like building with blocks – each concept builds upon the previous one. A solid understanding of the fundamentals is essential for tackling more complex problems in the future. So, let's dive in and build our vector algebra skills together!
Step 1: Scalar Multiplication
The first part of our journey involves scalar multiplication. This simply means multiplying a vector by a number (a scalar). We'll do this for each term in our equation for z:
- 7u = 7 * <1, 2, 3> = <7 * 1, 7 * 2, 7 * 3> = <7, 14, 21>
- 3v = 3 * <2, 2, -1> = <3 * 2, 3 * 2, 3 * -1> = <6, 6, -3>
- (1/2)w = (1/2) * <4, 0, -4> = <(1/2) * 4, (1/2) * 0, (1/2) * -4> = <2, 0, -2>
Scalar multiplication is a fundamental operation in vector algebra, and it's crucial to understand how it works. When you multiply a vector by a scalar, you're essentially scaling its magnitude (length) without changing its direction (unless the scalar is negative, in which case the direction is reversed). Each component of the vector is multiplied by the scalar, resulting in a new vector with components that are scaled proportionally. In our case, multiplying u by 7 stretches the vector by a factor of 7, multiplying v by 3 stretches it by a factor of 3, and multiplying w by 1/2 shrinks it by half. This operation is not just a mathematical abstraction; it has real-world applications. For example, in physics, if you have a force vector acting on an object and you double the force, you're essentially multiplying the force vector by the scalar 2. The resulting vector represents the new force, which has twice the magnitude but the same direction. Similarly, in computer graphics, scaling a 3D model involves multiplying the position vectors of its vertices by a scaling factor. This allows you to make the model larger or smaller without changing its shape. The concept of scalar multiplication extends beyond simple scaling. It also plays a crucial role in linear transformations, which are fundamental to many areas of mathematics and computer science. A linear transformation is a function that maps vectors to vectors in a way that preserves vector addition and scalar multiplication. This means that if you apply a linear transformation to a linear combination of vectors, the result will be the same linear combination of the transformed vectors. For example, consider a rotation in 2D space. Rotating a vector about the origin is a linear transformation because it preserves vector addition and scalar multiplication. Understanding scalar multiplication is essential for grasping the concept of linear transformations and their applications. So, as you perform the calculations in this step, keep in mind that you're not just manipulating numbers; you're manipulating vectors in a way that has a clear geometric interpretation. You're scaling their magnitudes and, if the scalar is negative, reversing their directions. This understanding will help you visualize the problem and develop a deeper intuition for vector algebra.
Step 2: Vector Addition and Subtraction
Now that we've handled the scalar multiplication, we can move on to vector addition and subtraction. Remember, we're trying to find:
z = 7u - 3v - (1/2)w
We've already calculated 7u, 3v, and (1/2)w. So, we can substitute those values into the equation:
z = <7, 14, 21> - <6, 6, -3> - <2, 0, -2>
To add or subtract vectors, we simply add or subtract their corresponding components. So:
z = <7 - 6 - 2, 14 - 6 - 0, 21 - (-3) - (-2)>
z = <7 - 6 - 2, 14 - 6, 21 + 3 + 2>
z = <-1, 8, 26>
And there you have it! We've found the vector z. Vector addition and subtraction are fundamental operations in linear algebra, and they have a clear geometric interpretation. When you add two vectors, you're essentially placing them head-to-tail and drawing a new vector from the tail of the first vector to the head of the second vector. This new vector represents the sum of the two original vectors. Similarly, when you subtract two vectors, you're essentially adding the first vector to the negative of the second vector. The negative of a vector has the same magnitude but the opposite direction. Vector addition and subtraction are used extensively in physics, engineering, and computer graphics. For example, in physics, you can use vector addition to find the resultant force acting on an object when multiple forces are applied. In engineering, you can use vector addition to analyze the forces acting on a structure. In computer graphics, you can use vector addition to move objects around in a 3D scene. The key to understanding vector addition and subtraction is to think of them as operations that combine vectors in a way that respects their magnitude and direction. When you add or subtract vectors, you're not just adding or subtracting numbers; you're adding or subtracting arrows that have a specific length and orientation. This geometric interpretation is crucial for developing a deeper intuition for vector algebra. In our case, we're subtracting 3v and (1/2)w from 7u. This means we're taking the vector 7u and then moving in the opposite direction of 3v and (1/2)w. The resulting vector z represents the final displacement after these movements. As you perform the calculations in this step, visualize the vectors as arrows in space and try to imagine how they are being combined. This will help you develop a better understanding of vector addition and subtraction and their applications.
Solution
So, after all the calculations, we've found that:
z = <-1, 8, 26>
That's our final answer! We successfully found the vector z by performing scalar multiplication and vector addition/subtraction. It's a pretty cool feeling when you solve a problem like this, isn't it? Finding the solution to a vector problem like this involves a systematic application of vector operations. We started by understanding the problem, then we performed scalar multiplication to scale the vectors u, v, and w by their respective coefficients. Finally, we added and subtracted the resulting vectors component-wise to obtain the vector z. This step-by-step approach is crucial for solving any mathematical problem, especially those involving multiple operations. By breaking down the problem into smaller, manageable steps, we can avoid errors and ensure that we arrive at the correct solution. The solution, z = <-1, 8, 26>, represents a vector in three-dimensional space. It has a specific magnitude and direction, which can be visualized as an arrow pointing from the origin to the point (-1, 8, 26). This vector is the result of combining the vectors u, v, and w in a specific way, as defined by the equation z = 7u - 3v - (1/2)w. Understanding the solution in the context of the problem is just as important as finding the solution itself. We've not only calculated the components of z, but we've also gained insight into how it relates to the original vectors u, v, and w. This understanding will help us apply these concepts to more complex problems in the future. Remember, mathematics is not just about finding answers; it's about understanding the underlying principles and how they can be applied to solve real-world problems. Vector algebra, in particular, is a powerful tool for modeling and solving problems in physics, engineering, and computer science. So, as you celebrate your success in finding the vector z, also take a moment to reflect on the process you used to get there. The skills and knowledge you've gained will serve you well in your future mathematical endeavors. And hey, if you ever get stuck on a similar problem, just remember the steps we took today: understand the problem, perform scalar multiplication, and then add or subtract the vectors. You got this!
Conclusion
Alright guys, we've successfully navigated the world of vectors and found our z! We took three vectors, scaled them, and combined them to create a new vector. This is a fundamental concept in linear algebra and has tons of applications in fields like physics, computer graphics, and more. I hope this step-by-step guide made the process clear and easy to follow. Keep practicing, and you'll be a vector pro in no time! The journey of solving this problem has reinforced several key concepts in vector algebra. We've seen how scalar multiplication affects the magnitude of a vector, and how vector addition and subtraction combine vectors to create new ones. We've also learned the importance of breaking down a complex problem into smaller, manageable steps. These are valuable skills that will serve you well in your future mathematical studies. But perhaps the most important takeaway is the understanding that mathematics is not just a collection of formulas and procedures; it's a way of thinking. By approaching problems with a systematic and logical mindset, we can unlock their solutions and gain a deeper appreciation for the beauty and power of mathematics. Vector algebra, in particular, is a fascinating field that connects abstract mathematical concepts to the physical world. Vectors can represent forces, velocities, displacements, and many other physical quantities. By manipulating vectors using algebraic operations, we can model and solve a wide range of problems in physics and engineering. For example, we can use vectors to analyze the motion of projectiles, the forces acting on a structure, or the flow of fluids. In computer graphics, vectors are used to represent the vertices of 3D models and the transformations that are applied to them. Scaling, rotation, and translation are all vector operations that are used to manipulate objects in a virtual world. So, as you continue your mathematical journey, remember that the concepts you're learning have real-world applications. Vector algebra is not just an abstract theory; it's a powerful tool that can be used to solve practical problems. And who knows, maybe one day you'll be using vector algebra to design a bridge, create a video game, or even explore the mysteries of the universe! Until then, keep practicing, keep exploring, and keep having fun with math!