Calculating Vector W From A Linear Combination Of V1, V2, And V3

Introduction

In the realm of linear algebra, the concept of combining vectors through scalar multiplication and addition is fundamental. This process, known as linear combination, allows us to express a new vector as a sum of scaled versions of other vectors. In this article, we will delve into the process of calculating a vector w as a linear combination of three given vectors: v1, v2, and v3. We'll explore the underlying principles and apply them to a specific example, providing a step-by-step guide to help you master this essential concept.

Problem Statement

Our objective is to determine the coordinates of the vector w, which is defined as a linear combination of the vectors v1, v2, and v3. We are given the following information:

• v1 = (-3, 0, 4)
• v2 = (21, -11, -9)
• v3 = (7, -8, 1)
• w = 1v1 - 1v2 + 2v3

To find the coordinates of w, we need to perform the scalar multiplications and vector additions as indicated in the expression for w. Let's break down the process step by step.

Step-by-Step Calculation

1. Scalar Multiplication

The first step involves multiplying each vector by its corresponding scalar coefficient. This operation scales the magnitude of the vector while preserving its direction (or reversing it if the scalar is negative).

• 1v1 = 1 * (-3, 0, 4) = (-3, 0, 4)
• -1v2 = -1 * (21, -11, -9) = (-21, 11, 9)
• 2v3 = 2 * (7, -8, 1) = (14, -16, 2)

2. Vector Addition

Next, we add the resulting scaled vectors together. Vector addition is performed component-wise, meaning we add the corresponding components of each vector.

w = 1v1 - 1v2 + 2v3 = (-3, 0, 4) + (-21, 11, 9) + (14, -16, 2)

To add the vectors, we sum their corresponding components:

• x-component: -3 + (-21) + 14 = -10
• y-component: 0 + 11 + (-16) = -5
• z-component: 4 + 9 + 2 = 15

Therefore, the vector w is:

w = (-10, -5, 15)

Analyzing the Result

By performing the linear combination, we have successfully determined the coordinates of vector w. The result, w = (-10, -5, 15), represents a new vector in three-dimensional space that is a combination of the original vectors v1, v2, and v3.

This process demonstrates the power of linear combinations in generating new vectors from existing ones. By varying the scalar coefficients, we can create a wide range of vectors that lie within the span of the original vectors. This concept is crucial in various applications, including computer graphics, physics, and engineering.

Alternative Solutions and Verification

While we have provided a detailed step-by-step calculation, it's always beneficial to explore alternative approaches and verify our results. One way to verify our answer is to use software or online calculators that perform vector operations. These tools can help confirm our calculations and provide additional insights.

Another approach is to visualize the vectors in three-dimensional space. Although it might be challenging to plot the vectors precisely, a rough sketch can help us understand the geometric relationship between v1, v2, v3, and w. This visualization can provide a qualitative check on our result.

Conclusion

In this article, we have successfully calculated the vector w as a linear combination of vectors v1, v2, and v3. By following a step-by-step approach, we performed scalar multiplication and vector addition to arrive at the solution: w = (-10, -5, 15). This process highlights the fundamental principles of linear algebra and the power of linear combinations in generating new vectors.

Understanding linear combinations is essential for various applications in mathematics, science, and engineering. By mastering this concept, you'll be well-equipped to tackle more complex problems involving vectors and vector spaces. Remember to practice and explore different examples to solidify your understanding. Keep exploring the fascinating world of linear algebra! In conclusion, we have determined the coordinates of vector w, which is a linear combination of vectors v1, v2, and v3. The calculated coordinates for vector w are (-10, -5, 15), derived from the expression w = 1v1 - 1v2 + 2v3, where v1=(-3, 0, 4), v2=(21, -11, -9), and v3=(7, -8, 1). This calculation involved scalar multiplication and vector addition, demonstrating a practical application of linear algebra principles. Understanding the concept of vector combination is crucial in various fields, including computer graphics and physics, as it allows for the generation of new vectors from existing ones. 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Answer

The correct answer is not explicitly provided in the text. Based on our calculations, the coordinates of vector w are (-10, -5, 15). Comparing this result with the given alternatives:

• A) (10, 16, -2) - Incorrect

The calculated answer (-10, -5, 15) does not match any of the provided alternatives. This could indicate an error in the alternatives provided or a misunderstanding in the problem statement. However, based on the given information and our calculations, (-10, -5, 15) is the correct answer.

Final Answer

The coordinates of the vector w are (-10, -5, 15).