Understanding Orthogonality

Oct 13, 2025 by windailys.com 92 views

Let $\vec{a}=\langle-1,4\rangle$ and $\vec{b}=\langle 5, k\rangle$ . Find $k$ so that $\vec{a}$ and $\vec{b}$ will be orthogonal (form a 90-degree angle).

Introduction

In mathematics, particularly in the field of vector geometry, two vectors are said to be orthogonal if they form a 90-degree angle with each other. This concept is crucial in various mathematical and scientific applications, including physics, engineering, and computer graphics. In this article, we will explore how to find the value of $k$ such that two given vectors, $\vec{a}$ and $\vec{b}$ , are orthogonal.

Understanding Orthogonality

For two vectors to be orthogonal, their dot product must be equal to zero. The dot product of two vectors $\vec{a} = \langle a_1, a_2 \rangle$ and $\vec{b} = \langle b_1, b_2 \rangle$ is given by the formula:

\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2

In our case, we have $\vec{a} = \langle -1, 4 \rangle$ and $\vec{b} = \langle 5, k \rangle$ . To find the value of $k$ such that $\vec{a}$ and $\vec{b}$ are orthogonal, we need to set up the equation:

\vec{a} \cdot \vec{b} = (-1)(5) + (4)(k) = 0

Solving for $k$

Now, let's solve the equation for $k$ :

-5 + 4k = 0

To isolate $k$ , we can add 5 to both sides of the equation:

4k = 5

Next, we can divide both sides of the equation by 4 to solve for $k$ :

k = \frac{5}{4}

Conclusion

In this article, we have found the value of $k$ such that the vectors $\vec{a} = \langle -1, 4 \rangle$ and $\vec{b} = \langle 5, k \rangle$ are orthogonal. The value of $k$ is $\frac{5}{4}$ . This result is crucial in various mathematical and scientific applications, including physics, engineering, and computer graphics.

Applications of Orthogonality

Orthogonality has numerous applications in various fields, including:

Physics: In physics, orthogonality is used to describe the relationship between vectors in three-dimensional space. For example, the dot product of two vectors is used to calculate the work done by a force on an object.
Engineering: In engineering, orthogonality is used to design and analyze systems, such as electrical circuits and mechanical systems.
Computer Graphics: In computer graphics, orthogonality is used to create 3D models and animations.

Final Thoughts

In conclusion, orthogonality is a fundamental concept in mathematics and has numerous applications in various fields. By understanding how to find the value of $k$ such that two vectors are orthogonal, we can apply this concept to solve real-world problems.
Q&A: Orthogonality of Vectors

Introduction

In our previous article, we explored how to find the value of $k$ such that two given vectors, $\vec{a}$ and $\vec{b}$ , are orthogonal. In this article, we will answer some frequently asked questions about orthogonality of vectors.

Q: What is the dot product of two vectors?

A: The dot product of two vectors $\vec{a} = \langle a_1, a_2 \rangle$ and $\vec{b} = \langle b_1, b_2 \rangle$ is given by the formula:

\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2

Q: How do I know if two vectors are orthogonal?

A: Two vectors are orthogonal if their dot product is equal to zero. In other words, if $\vec{a} \cdot \vec{b} = 0$ , then the vectors are orthogonal.

Q: Can two vectors be orthogonal if they have different magnitudes?

A: Yes, two vectors can be orthogonal even if they have different magnitudes. For example, the vectors $\vec{a} = \langle 1, 0 \rangle$ and $\vec{b} = \langle 0, 1 \rangle$ are orthogonal, even though they have different magnitudes.

Q: Can two vectors be orthogonal if they are parallel?

A: No, two vectors cannot be orthogonal if they are parallel. If two vectors are parallel, their dot product is equal to the product of their magnitudes, which is not equal to zero.

Q: How do I find the value of $k$ such that two vectors are orthogonal?

A: To find the value of $k$ such that two vectors are orthogonal, you can set up the equation:

\vec{a} \cdot \vec{b} = 0

and solve for $k$ .

Q: What is the significance of orthogonality in physics?

A: Orthogonality is crucial in physics because it allows us to describe the relationship between vectors in three-dimensional space. For example, the dot product of two vectors is used to calculate the work done by a force on an object.

Q: Can orthogonality be used in other fields besides physics?

A: Yes, orthogonality can be used in other fields besides physics, such as engineering and computer graphics. In engineering, orthogonality is used to design and analyze systems, such as electrical circuits and mechanical systems. In computer graphics, orthogonality is used to create 3D models and animations.

Q: How do I apply orthogonality in real-world problems?

A: To apply orthogonality in real-world problems, you can use the concept of orthogonality to solve problems that involve vectors. For example, you can use orthogonality to calculate the work done by a force on an object, or to design and analyze systems.

Conclusion

In this article, we have answered some frequently asked questions about orthogonality of vectors. We hope that this article has provided you with a better understanding of the concept of orthogonality and how it can be applied in real-world problems.

Additional Resources

If you are interested in learning more about orthogonality, 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: "MathWorld" and "Wolfram Alpha".

We hope that this article has been helpful in your understanding of orthogonality. If you have any further questions, please don't hesitate to ask.