Multiply each vector by its corresponding scalar

Oct 13, 2025 by windailys.com 53 views

Introduction

In vector mathematics, we often need to perform operations on vectors to find new vectors. One common operation is finding the result of a linear combination of vectors. In this article, we will explore how to find the result of a linear combination of two given vectors, specifically the expression $ $-5a + 2b$ $. We will use the given vectors $$a =\langle 5,-3\rangle$$ and $$b =\langle 4,1\rangle$$ to demonstrate the process.

Understanding Vector Operations

Before we dive into the problem, let's review the basics of vector operations. Vectors are mathematical objects that have both magnitude (length) and direction. They can be represented graphically as arrows in a coordinate system. Vector operations involve combining or manipulating vectors to find new vectors.

There are several types of vector operations, including:

Scalar multiplication: This involves multiplying a vector by a scalar (a number) to change its magnitude.
Vector addition: This involves adding two or more vectors to find a new vector.
Linear combination: This involves combining vectors using scalar multiplication and vector addition.

Finding the Result of a Linear Combination

To find the result of a linear combination of vectors, we need to follow these steps:

Multiply each vector by its corresponding scalar: In the expression $ $-5a + 2b$ $, we need to multiply vector $$a$$ by the scalar $$-5$$ and vector $$b$$ by the scalar $$2$$.
Add the resulting vectors: After multiplying each vector by its corresponding scalar, we need to add the resulting vectors to find the final result.

Calculating the Result

Now, let's calculate the result of the linear combination $ $-5a + 2b$ $ using the given vectors $$a =\langle 5,-3\rangle$$ and $$b =\langle 4,1\rangle$$.

Step 1: Multiply Each Vector by Its Corresponding Scalar

To multiply each vector by its corresponding scalar, we need to multiply each component of the vector by the scalar.

For vector $$a =\langle 5,-3\rangle$$, we multiply each component by the scalar $$-5$$:
- $-5 \cdot 5 = -25$
- $-5 \cdot (-3) = 15$
So, $$-5a = \langle -25, 15\rangle$$.
For vector $$b =\langle 4,1\rangle$$, we multiply each component by the scalar $$2$$:
- $2 \cdot 4 = 8$
- $2 \cdot 1 = 2$
So, $$2b = \langle 8, 2\rangle$$.

Step 2: Add the Resulting Vectors

Now that we have multiplied each vector by its corresponding scalar, we need to add the resulting vectors to find the final result.

To add the vectors $$-5a = \langle -25, 15\rangle$$ and $$2b = \langle 8, 2\rangle$$, we add their corresponding components:
- $-25 + 8 = -17$
- $15 + 2 = 17$
So, the final result is $$-5a + 2b = \langle -17, 17\rangle$$.

Conclusion

In this article, we explored how to find the result of a linear combination of two given vectors, specifically the expression $ $-5a + 2b$ $. We used the given vectors $$a =\langle 5,-3\rangle$$ and $$b =\langle 4,1\rangle$$ to demonstrate the process. By following the steps of scalar multiplication and vector addition, we found the final result to be $$-5a + 2b = \langle -17, 17\rangle$$.

Applications of Vector Operations

Vector operations have numerous applications in mathematics, physics, engineering, and computer science. Some examples include:

Linear algebra: Vector operations are used to solve systems of linear equations and find the inverse of matrices.
Physics: Vector operations are used to describe the motion of objects in space and time.
Computer graphics: Vector operations are used to create 3D models and animations.
Machine learning: Vector operations are used to train neural networks and perform data analysis.

Final Thoughts

Vector operations are a fundamental concept in mathematics and have numerous applications in various fields. By understanding how to perform vector operations, we can solve complex problems and make new discoveries. In this article, we explored how to find the result of a linear combination of two given vectors, specifically the expression $ $-5a + 2b$ $. We hope that this article has provided a clear and concise explanation of the process and has inspired readers to explore the world of vector mathematics.

Introduction

In our previous article, we explored how to find the result of a linear combination of two given vectors, specifically the expression $ $-5a + 2b$ $. We used the given vectors $$a =\langle 5,-3\rangle$$ and $$b =\langle 4,1\rangle$$ to demonstrate the process. In this article, we will answer some frequently asked questions about vector operations and provide additional examples to help solidify your understanding.

Q&A

Q: What is the difference between scalar multiplication and vector addition?

A: Scalar multiplication involves multiplying a vector by a scalar (a number) to change its magnitude, while vector addition involves adding two or more vectors to find a new vector.

Q: How do I perform scalar multiplication on a vector?

A: To perform scalar multiplication on a vector, you multiply each component of the vector by the scalar. For example, if you have a vector $$a =\langle 5,-3\rangle$$ and you want to multiply it by the scalar $$-5$$, you would get $$-5a = \langle -25, 15\rangle$$.

Q: How do I perform vector addition?

A: To perform vector addition, you add the corresponding components of the vectors. For example, if you have two vectors $$a =\langle 5,-3\rangle$$ and $$b =\langle 4,1\rangle$$, you would add their corresponding components to get $$a + b = \langle 9, -2\rangle$$.

Q: What is the result of the expression $ $-3a + 2b$ $?

A: To find the result of the expression $ $-3a + 2b$ $, we need to multiply each vector by its corresponding scalar and then add the resulting vectors. Using the given vectors $$a =\langle 5,-3\rangle$$ and $$b =\langle 4,1\rangle$$, we get:

$-3a = \langle -15, 9\rangle$
$2b = \langle 8, 2\rangle$
$-3a + 2b = \langle -7, 11\rangle$

Q: What is the result of the expression $ $2a - 3b$ $?

A: To find the result of the expression $ $2a - 3b$ $, we need to multiply each vector by its corresponding scalar and then subtract the resulting vectors. Using the given vectors $$a =\langle 5,-3\rangle$$ and $$b =\langle 4,1\rangle$$, we get:

$2a = \langle 10, -6\rangle$
$-3b = \langle -12, -3\rangle$
$2a - 3b = \langle -2, -9\rangle$

Q: Can I add or subtract vectors with different magnitudes?

A: Yes, you can add or subtract vectors with different magnitudes. However, the resulting vector will have a magnitude that is the sum or difference of the magnitudes of the original vectors.

Q: Can I multiply a vector by a scalar that is a fraction?

A: Yes, you can multiply a vector by a scalar that is a fraction. For example, if you have a vector $$a =\langle 5,-3\rangle$$ and you want to multiply it by the scalar $$\frac{1}{2}$$, you would get $$\frac{1}{2}a = \langle \frac{5}{2}, -\frac{3}{2}\rangle$$.

Conclusion

In this article, we answered some frequently asked questions about vector operations and provided additional examples to help solidify your understanding. We hope that this article has provided a clear and concise explanation of the process and has inspired readers to explore the world of vector mathematics.

Final Thoughts

Vector operations are a fundamental concept in mathematics and have numerous applications in various fields. By understanding how to perform vector operations, we can solve complex problems and make new discoveries. We hope that this article has provided a valuable resource for readers who are interested in learning more about vector mathematics.

Additional Resources

If you are interested in learning more about vector mathematics, we recommend checking out the following resources:

Linear Algebra: A comprehensive textbook on linear algebra that covers vector operations and other topics.
Vector Calculus: A textbook on vector calculus that covers topics such as vector fields and line integrals.
Mathematics Online Courses: Online courses on mathematics that cover vector operations and other topics.

Frequently Asked Questions

Q: What is the difference between a vector and a scalar? A: A vector is a mathematical object that has both magnitude (length) and direction, while a scalar is a number that has no direction.
Q: How do I perform vector operations? A: To perform vector operations, you need to follow the rules of scalar multiplication and vector addition.
Q: Can I add or subtract vectors with different magnitudes? A: Yes, you can add or subtract vectors with different magnitudes. However, the resulting vector will have a magnitude that is the sum or difference of the magnitudes of the original vectors.
Q: Can I multiply a vector by a scalar that is a fraction? A: Yes, you can multiply a vector by a scalar that is a fraction. For example, if you have a vector $$a =\langle 5,-3\rangle$$ and you want to multiply it by the scalar $$\frac{1}{2}$$, you would get $$\frac{1}{2}a = \langle \frac{5}{2}, -\frac{3}{2}\rangle$$.

Glossary

Vector: A mathematical object that has both magnitude (length) and direction.
Scalar: A number that has no direction.
Scalar Multiplication: The process of multiplying a vector by a scalar to change its magnitude.
Vector Addition: The process of adding two or more vectors to find a new vector.
Linear Combination: The process of combining vectors using scalar multiplication and vector addition.