Let { \overrightarrow{DE}$}$ Be The Vector With Initial Point { D(8,6)$}$ And Terminal Point { E(-1,5)$}$. Write { \overrightarrow{DE}$}$ As A Linear Combination Of The Vectors { \mathbf{i}$}$ And

Introduction

In vector operations, we often need to express a given vector as a linear combination of other vectors. This is a fundamental concept in mathematics, particularly in linear algebra. In this article, we will explore how to write a vector as a linear combination of the standard unit vectors i and j.

The Problem

Let $\overrightarrow{DE}$ be the vector with initial point $D(8,6)$ and terminal point $E(-1,5)$. Our goal is to write $\overrightarrow{DE}$ as a linear combination of the vectors $\mathbf{i}$ and $\mathbf{j}$.

Understanding the Vectors

Before we proceed, let's understand the vectors involved. The vector $\overrightarrow{DE}$ is a two-dimensional vector with components $x$ and $y$. The standard unit vectors $\mathbf{i}$ and $\mathbf{j}$ are also two-dimensional vectors with components $1$ and $0$, respectively.

Writing the Vector as a Linear Combination

To write $\overrightarrow{DE}$ as a linear combination of $\mathbf{i}$ and $\mathbf{j}$, we need to find the coefficients $a$ and $b$ such that:

$$\overrightarrow{DE} = a\mathbf{i} + b\mathbf{j}$$

We can find the coefficients $a$ and $b$ by using the components of the vector $\overrightarrow{DE}$. The $x$-component of $\overrightarrow{DE}$ is given by:

$$x = -1 - 8 = -9$$

The $y$-component of $\overrightarrow{DE}$ is given by:

$$y = 5 - 6 = -1$$

Now, we can write the vector $\overrightarrow{DE}$ as a linear combination of $\mathbf{i}$ and $\mathbf{j}$:

$$\overrightarrow{DE} = -9\mathbf{i} - 1\mathbf{j}$$

Interpretation

In this example, we have written the vector $\overrightarrow{DE}$ as a linear combination of the standard unit vectors $\mathbf{i}$ and $\mathbf{j}$. The coefficients $a$ and $b$ represent the $x$- and $y$-components of the vector $\overrightarrow{DE}$, respectively.

Conclusion

In this article, we have explored how to write a vector as a linear combination of the standard unit vectors $\mathbf{i}$ and $\mathbf{j}$. We have used the components of the vector to find the coefficients $a$ and $b$, and have written the vector as a linear combination of $\mathbf{i}$ and $\mathbf{j}$. This is a fundamental concept in mathematics, particularly in linear algebra, and is used extensively in various fields such as physics, engineering, and computer science.

Applications

The concept of writing a vector as a linear combination of other vectors has numerous applications in various fields. Some of the applications include:

• Physics: In physics, vectors are used to represent physical quantities such as displacement, velocity, and acceleration. Writing a vector as a linear combination of other vectors is essential in solving problems involving motion and forces.
• Engineering: In engineering, vectors are used to represent physical quantities such as displacement, velocity, and acceleration. Writing a vector as a linear combination of other vectors is essential in solving problems involving motion and forces.
• Computer Science: In computer science, vectors are used to represent data structures such as arrays and lists. Writing a vector as a linear combination of other vectors is essential in solving problems involving data structures and algorithms.

Example Problems

Here are some example problems that involve writing a vector as a linear combination of other vectors:

• Problem 1: Let $\overrightarrow{AB}$ be the vector with initial point $A(2,3)$ and terminal point $B(5,7)$. Write $\overrightarrow{AB}$ as a linear combination of the vectors $\mathbf{i}$ and $\mathbf{j}$.
• Problem 2: Let $\overrightarrow{CD}$ be the vector with initial point $C(1,2)$ and terminal point $D(3,4)$. Write $\overrightarrow{CD}$ as a linear combination of the vectors $\mathbf{i}$ and $\mathbf{j}$.

Solutions

Here are the solutions to the example problems:

• Problem 1: $\overrightarrow{AB} = 3\mathbf{i} + 4\mathbf{j}$
• Problem 2: $\overrightarrow{CD} = 2\mathbf{i} + 2\mathbf{j}$

Conclusion

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Introduction

In our previous article, we explored how to write a vector as a linear combination of the standard unit vectors i and j. In this article, we will answer some frequently asked questions (FAQs) related to vector operations and writing a vector as a linear combination.

Q&A

Q: What is a vector?

A: A vector is a mathematical object that has both magnitude (length) and direction. It is often represented graphically as an arrow in a coordinate system.

Q: What are the standard unit vectors i and j?

A: The standard unit vectors i and j are two-dimensional vectors with components 1 and 0, respectively. They are used as a basis for representing other vectors in a coordinate system.

Q: How do I write a vector as a linear combination of i and j?

A: To write a vector as a linear combination of i and j, you need to find the coefficients a and b such that:

$$\overrightarrow{DE} = a\mathbf{i} + b\mathbf{j}$$

You can find the coefficients a and b by using the components of the vector DE.

Q: What are the components of a vector?

A: The components of a vector are the values that represent the magnitude and direction of the vector. For a two-dimensional vector, the components are typically represented as x and y.

Q: How do I find the components of a vector?

A: To find the components of a vector, you need to know the initial and terminal points of the vector. The x-component of the vector is given by the difference between the x-coordinates of the terminal and initial points, while the y-component is given by the difference between the y-coordinates of the terminal and initial points.

Q: What is the difference between a vector and a scalar?

A: A vector is a mathematical object that has both magnitude and direction, while a scalar is a mathematical object that has only magnitude. Scalars are often represented as numbers, while vectors are represented as arrows in a coordinate system.

Q: Can I write a vector as a linear combination of more than two vectors?

A: Yes, you can write a vector as a linear combination of more than two vectors. However, this is typically done in higher-dimensional spaces, such as three-dimensional or four-dimensional spaces.

Q: What are some common applications of writing a vector as a linear combination?

A: Writing a vector as a linear combination has numerous applications in various fields, including physics, engineering, and computer science. Some common applications include:

• Physics: Writing a vector as a linear combination is essential in solving problems involving motion and forces.
• Engineering: Writing a vector as a linear combination is essential in solving problems involving motion and forces.
• Computer Science: Writing a vector as a linear combination is essential in solving problems involving data structures and algorithms.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to vector operations and writing a vector as a linear combination. We hope that this article has provided you with a better understanding of vector operations and how to write a vector as a linear combination.

Additional Resources

For more information on vector operations and writing a vector as a linear combination, we recommend the following resources:

• Textbooks: "Linear Algebra and Its Applications" by Gilbert Strang and "Vector Calculus" by Michael Spivak
• Online Courses: "Linear Algebra" by MIT OpenCourseWare and "Vector Calculus" by Khan Academy
• Websites: "Math Is Fun" and "Khan Academy"

Practice Problems

Here are some practice problems to help you reinforce your understanding of vector operations and writing a vector as a linear combination:

• Problem 1: Let DE be the vector with initial point D(2,3) and terminal point E(5,7). Write DE as a linear combination of i and j.
• Problem 2: Let CD be the vector with initial point C(1,2) and terminal point D(3,4). Write CD as a linear combination of i and j.

Solutions

Here are the solutions to the practice problems:

• Problem 1: DE = 3i + 4j
• Problem 2: CD = 2i + 2j