Let { \vec{a}$}$ And { \vec{b}$}$ Be Vectors Such That { |\vec{a}| = 4$}$ And { |2 \vec{a} - \vec{b}| = |\vec{a} + \vec{b}|$}$. Which One Of The Following Is Equal To { \vec{a} \cdot \vec{b}$}$?A. 12

In mathematics, vectors are an essential concept used to represent quantities with both magnitude and direction. The dot product of two vectors is a fundamental operation that helps in finding the angle between them. In this article, we will explore the concept of dot product and its application in solving a problem involving vectors.

What is the Dot Product?

The dot product of two vectors ${\vec{a}}$ and ${\vec{b}}$ is denoted by ${\vec{a} \cdot \vec{b}}$ and is defined as the product of the magnitudes of the vectors and the cosine of the angle between them. Mathematically, it can be represented as:

${\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta}$

where ${|\vec{a}|}$ and ${|\vec{b}|}$ are the magnitudes of the vectors and ${\theta}$ is the angle between them.

Problem Statement

Let ${\vec{a}}$ and ${\vec{b}}$ be vectors such that ${|\vec{a}| = 4}$ and ${|2 \vec{a} - \vec{b}| = |\vec{a} + \vec{b}|}$. We need to find the value of ${\vec{a} \cdot \vec{b}}$.

Step 1: Understanding the Given Information

We are given that ${|\vec{a}| = 4}$, which means the magnitude of vector ${\vec{a}}$ is 4 units. We are also given that ${|2 \vec{a} - \vec{b}| = |\vec{a} + \vec{b}|}$, which means the magnitude of the vector ${2 \vec{a} - \vec{b}}$ is equal to the magnitude of the vector ${\vec{a} + \vec{b}}$.

Step 2: Squaring Both Sides of the Equation

To simplify the equation, we can square both sides of the equation:

${|2 \vec{a} - \vec{b}|^2 = |\vec{a} + \vec{b}|^2}$

Expanding the squared terms, we get:

${(2 \vec{a} - \vec{b}) \cdot (2 \vec{a} - \vec{b}) = (\vec{a} + \vec{b}) \cdot (\vec{a} + \vec{b})}$

Step 3: Expanding the Dot Product

Expanding the dot product on both sides of the equation, we get:

${4 |\vec{a}|^2 - 4 \vec{a} \cdot \vec{b} + |\vec{b}|^2 = |\vec{a}|^2 + 2 \vec{a} \cdot \vec{b} + |\vec{b}|^2}$

Step 4: Simplifying the Equation

Simplifying the equation, we get:

${4 |\vec{a}|^2 - 4 \vec{a} \cdot \vec{b} = |\vec{a}|^2 + 2 \vec{a} \cdot \vec{b}}$

Substituting the value of ${|\vec{a}| = 4}$, we get:

${16 - 4 \vec{a} \cdot \vec{b} = 4 + 2 \vec{a} \cdot \vec{b}}$

Step 5: Solving for ${\vec{a} \cdot \vec{b}}$

Solving for ${\vec{a} \cdot \vec{b}}$, we get:

${16 - 4 = 4 + 2 \vec{a} \cdot \vec{b}}$

${12 = 2 \vec{a} \cdot \vec{b}}$

${\vec{a} \cdot \vec{b} = 6}$

Therefore, the value of ${\vec{a} \cdot \vec{b}}$ is 6.

Conclusion

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In this article, we will address some of the frequently asked questions related to the concept of dot product and its application in solving problems involving vectors.

Q: What is the dot product of two vectors?

A: The dot product of two vectors ${\vec{a}}$ and ${\vec{b}}$ is denoted by ${\vec{a} \cdot \vec{b}}$ and is defined as the product of the magnitudes of the vectors and the cosine of the angle between them.

Q: How is the dot product used in solving problems involving vectors?

A: The dot product is used to find the angle between two vectors, the projection of one vector onto another, and the work done by a force on an object.

Q: What is the formula for the dot product?

A: The formula for the dot product is:

${\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta}$

where ${|\vec{a}|}$ and ${|\vec{b}|}$ are the magnitudes of the vectors and ${\theta}$ is the angle between them.

Q: How do you find the magnitude of a vector?

A: The magnitude of a vector ${\vec{a}}$ is found using the formula:

${|\vec{a}| = \sqrt{x^2 + y^2 + z^2}}$

where ${x}$, ${y}$, and ${z}$ are the components of the vector.

Q: What is the difference between the dot product and the cross product?

A: The dot product is used to find the angle between two vectors, while the cross product is used to find the area of the parallelogram formed by the two vectors.

Q: Can you give an example of how to use the dot product to solve a problem?

A: Yes, consider the problem of finding the angle between two vectors ${\vec{a}}$ and ${\vec{b}}$. We can use the dot product formula to find the angle:

${\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta}$

Rearranging the formula to solve for ${\theta}$, we get:

${\cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}}$

Q: What are some common applications of the dot product?

A: The dot product has many applications in physics, engineering, and computer science, including:

• Finding the angle between two vectors
• Finding the projection of one vector onto another
• Finding the work done by a force on an object
• Finding the energy of a system

Q: Can you give an example of how to use the dot product to find the work done by a force on an object?

A: Yes, consider the problem of finding the work done by a force ${\vec{F}}$ on an object that is moving along a path defined by the vector ${\vec{r}}$. We can use the dot product formula to find the work done:

${W = \vec{F} \cdot \vec{r}}$

This formula is used in many real-world applications, including the calculation of the energy required to move an object from one point to another.

Conclusion

In this article, we have addressed some of the frequently asked questions related to the concept of dot product and its application in solving problems involving vectors. We hope that this article has provided a clear understanding of the dot product and its many applications.