Find A ⋅ B A \cdot B A ⋅ B If A = ⟨ 4 , 3 ⟩ A = \langle 4,3 \rangle A = ⟨ 4 , 3 ⟩ And B = ⟨ 4 , 5 ⟩ B = \langle 4,5 \rangle B = ⟨ 4 , 5 ⟩ .A. 31 B. ( 8 , 8 (8,8 ( 8 , 8 ] C. ⟨ 16 , 15 ⟩ \langle 16,15 \rangle ⟨ 16 , 15 ⟩ D. -1 Please Select The Best Answer From The Choices Provided.
Introduction
In mathematics, the dot product, also known as the scalar product, is a way to multiply two vectors together to get a scalar value. It is a fundamental concept in linear algebra and is used in many areas of mathematics and physics. In this article, we will explore how to find the dot product of two vectors and apply this concept to a specific problem.
What is the Dot Product?
The dot product of two vectors a and b is denoted by a · b and is defined as the sum of the products of the corresponding components of the two vectors. In other words, if a = 〈a1, a2〉 and b = 〈b1, b2〉, then the dot product a · b is given by:
a · b = a1b1 + a2b2
How to Find the Dot Product
To find the dot product of two vectors, we simply multiply the corresponding components of the two vectors and add the results together. Let's consider an example to illustrate this concept.
Example: Finding the Dot Product of Two Vectors
Suppose we have two vectors a = 〈4, 3〉 and b = 〈4, 5〉. To find the dot product of these two vectors, we multiply the corresponding components and add the results together:
a · b = (4)(4) + (3)(5) a · b = 16 + 15 a · b = 31
Therefore, the dot product of the two vectors a and b is 31.
Applying the Concept to a Problem
Now that we have a good understanding of how to find the dot product of two vectors, let's apply this concept to the problem presented in the question. We are given two vectors a = 〈4, 3〉 and b = 〈4, 5〉, and we are asked to find the dot product of these two vectors.
Using the formula for the dot product, we can find the dot product of the two vectors as follows:
a · b = (4)(4) + (3)(5) a · b = 16 + 15 a · b = 31
Therefore, the dot product of the two vectors a and b is 31.
Conclusion
In this article, we have explored the concept of the dot product of two vectors and applied this concept to a specific problem. We have seen that the dot product of two vectors is a scalar value that is obtained by multiplying the corresponding components of the two vectors and adding the results together. We have also seen that the dot product of two vectors can be used to solve problems in mathematics and physics.
Answer
Based on the calculation above, the correct answer is:
A. 31
Final Thoughts
Q: What is the dot product of two vectors?
A: The dot product of two vectors a and b is a scalar value that is obtained by multiplying the corresponding components of the two vectors and adding the results together. It is denoted by a · b.
Q: How do I find the dot product of two vectors?
A: To find the dot product of two vectors, you simply multiply the corresponding components of the two vectors and add the results together. For example, if a = 〈a1, a2〉 and b = 〈b1, b2〉, then the dot product a · b is given by:
a · b = a1b1 + a2b2
Q: What are the units of the dot product of two vectors?
A: The units of the dot product of two vectors are the same as the units of the scalar value that is obtained by multiplying the corresponding components of the two vectors and adding the results together.
Q: Can the dot product of two vectors be negative?
A: Yes, the dot product of two vectors can be negative. This occurs when the corresponding components of the two vectors have opposite signs.
Q: Can the dot product of two vectors be zero?
A: Yes, the dot product of two vectors can be zero. This occurs when the corresponding components of the two vectors are both zero.
Q: What is the geometric interpretation of the dot product of two vectors?
A: The dot product of two vectors can be interpreted as the product of the magnitudes of the two vectors and the cosine of the angle between them. This is known as the geometric interpretation of the dot product.
Q: How is the dot product of two vectors used in physics?
A: The dot product of two vectors is used in physics to calculate the work done by a force on an object, the power of a force, and the torque of a force.
Q: Can the dot product of two vectors be used to find the angle between two vectors?
A: Yes, the dot product of two vectors can be used to find the angle between two vectors. This is known as the cosine law.
Q: What are some common applications of the dot product of two vectors?
A: Some common applications of the dot product of two vectors include:
- Calculating the work done by a force on an object
- Calculating the power of a force
- Calculating the torque of a force
- Finding the angle between two vectors
- Calculating the magnitude of a vector
Q: Can the dot product of two vectors be used in computer graphics?
A: Yes, the dot product of two vectors can be used in computer graphics to perform tasks such as:
- Calculating the normal of a surface
- Calculating the reflection of a light ray
- Calculating the refraction of a light ray
Q: Can the dot product of two vectors be used in machine learning?
A: Yes, the dot product of two vectors can be used in machine learning to perform tasks such as:
- Calculating the similarity between two vectors
- Calculating the distance between two vectors
- Calculating the cosine similarity between two vectors
Conclusion
In this article, we have answered some frequently asked questions about the dot product of two vectors. We have seen that the dot product of two vectors is a scalar value that is obtained by multiplying the corresponding components of the two vectors and adding the results together. We have also seen that the dot product of two vectors has many applications in physics, computer graphics, and machine learning.