If $\vec{a} + \vec{b} + \vec{c} = 0$, $|\vec{a}| = 5$, $|\vec{b}| = 4$, And $|\vec{c}| = 3$, Then Find $\vec{b} \cdot \vec{c}$.

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Introduction

In this article, we will explore the concept of vectors and their properties. We will use the given information to find the dot product of two vectors, $\vec{b}$ and $\vec{c}$. The dot product is a fundamental concept in mathematics and physics, and it has numerous applications in various fields.

Understanding Vectors

A vector is a mathematical object that has both magnitude and direction. It can be represented graphically as an arrow in a coordinate system. The magnitude of a vector is its length, and the direction is the angle it makes with the positive x-axis.

Given Information

We are given that $\vec{a} + \vec{b} + \vec{c} = 0$, $|\vec{a}| = 5$, $|\vec{b}| = 4$, and $|\vec{c}| = 3$. This means that the sum of the three vectors is equal to the zero vector, and the magnitudes of the vectors are given.

Finding the Dot Product

The dot product of two vectors, $\vec{u}$ and $\vec{v}$, is defined as:

$$\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta$$

where $\theta$ is the angle between the two vectors.

We can use the given information to find the dot product of $\vec{b}$ and $\vec{c}$. Since $\vec{a} + \vec{b} + \vec{c} = 0$, we can write:

$$\vec{a} = -\vec{b} - \vec{c}$$

Taking the magnitude of both sides, we get:

$$|\vec{a}| = |-\vec{b} - \vec{c}|$$

Using the triangle inequality, we can write:

$$|\vec{a}| \leq |\vec{b}| + |\vec{c}|$$

Substituting the given values, we get:

$$5 \leq 4 + 3$$

This is a true statement, so we can conclude that the triangle inequality holds.

Using the Dot Product Formula

Now, we can use the dot product formula to find the dot product of $\vec{b}$ and $\vec{c}$. We can write:

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

where $\theta$ is the angle between $\vec{b}$ and $\vec{c}$.

Since $\vec{a} + \vec{b} + \vec{c} = 0$, we can write:

$$\vec{a} = -\vec{b} - \vec{c}$$

Taking the dot product of both sides with $\vec{b}$, we get:

$$\vec{a} \cdot \vec{b} = -\vec{b} \cdot \vec{b} - \vec{b} \cdot \vec{c}$$

Using the dot product formula, we can write:

$$|\vec{a}| |\vec{b}| \cos \theta = -|\vec{b}|^2 - \vec{b} \cdot \vec{c}$$

Substituting the given values, we get:

$$5 \cdot 4 \cos \theta = -4^2 - \vec{b} \cdot \vec{c}$$

Simplifying, we get:

$$20 \cos \theta = -16 - \vec{b} \cdot \vec{c}$$

Now, we can use the fact that $\vec{a} + \vec{b} + \vec{c} = 0$ to find the dot product of $\vec{a}$ and $\vec{b}$. We can write:

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

Substituting the given values, we get:

$$5 \cdot 4 \cos \theta = 20 \cos \theta$$

Simplifying, we get:

$$20 \cos \theta = 20 \cos \theta$$

This is a true statement, so we can conclude that the dot product of $\vec{a}$ and $\vec{b}$ is equal to the dot product of $\vec{b}$ and $\vec{c}$.

**Finding the Dot Product of $\vec{b}$ and $\vec{c}$$

Now, we can use the fact that the dot product of $\vec{a}$ and $\vec{b}$ is equal to the dot product of $\vec{b}$ and $\vec{c}$ to find the dot product of $\vec{b}$ and $\vec{c}$. We can write:

$$\vec{b} \cdot \vec{c} = -16 - \vec{b} \cdot \vec{c}$$

Simplifying, we get:

$$2 \vec{b} \cdot \vec{c} = -16$$

Dividing both sides by 2, we get:

$$\vec{b} \cdot \vec{c} = -8$$

Therefore, the dot product of $\vec{b}$ and $\vec{c}$ is -8.

Conclusion

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Q: What is the dot product of two vectors?

A: The dot product of two vectors, $\vec{u}$ and $\vec{v}$, is defined as:

$$\vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos \theta$$

where $\theta$ is the angle between the two vectors.

Q: How do we find the dot product of $\vec{b}$ and $\vec{c}$?

A: We can use the given information to find the dot product of $\vec{b}$ and $\vec{c}$. Since $\vec{a} + \vec{b} + \vec{c} = 0$, we can write:

$$\vec{a} = -\vec{b} - \vec{c}$$

Taking the dot product of both sides with $\vec{b}$, we get:

$$\vec{a} \cdot \vec{b} = -\vec{b} \cdot \vec{b} - \vec{b} \cdot \vec{c}$$

Using the dot product formula, we can write:

$$|\vec{a}| |\vec{b}| \cos \theta = -|\vec{b}|^2 - \vec{b} \cdot \vec{c}$$

Substituting the given values, we get:

$$5 \cdot 4 \cos \theta = -4^2 - \vec{b} \cdot \vec{c}$$

Simplifying, we get:

$$20 \cos \theta = -16 - \vec{b} \cdot \vec{c}$$

Q: How do we find the dot product of $\vec{a}$ and $\vec{b}$?

A: We can use the fact that $\vec{a} + \vec{b} + \vec{c} = 0$ to find the dot product of $\vec{a}$ and $\vec{b}$. We can write:

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

Substituting the given values, we get:

$$5 \cdot 4 \cos \theta = 20 \cos \theta$$

Simplifying, we get:

$$20 \cos \theta = 20 \cos \theta$$

This is a true statement, so we can conclude that the dot product of $\vec{a}$ and $\vec{b}$ is equal to the dot product of $\vec{b}$ and $\vec{c}$.

Q: How do we find the dot product of $\vec{b}$ and $\vec{c}$?

A: We can use the fact that the dot product of $\vec{a}$ and $\vec{b}$ is equal to the dot product of $\vec{b}$ and $\vec{c}$ to find the dot product of $\vec{b}$ and $\vec{c}$. We can write:

$$\vec{b} \cdot \vec{c} = -16 - \vec{b} \cdot \vec{c}$$

Simplifying, we get:

$$2 \vec{b} \cdot \vec{c} = -16$$

Dividing both sides by 2, we get:

$$\vec{b} \cdot \vec{c} = -8$$

Therefore, the dot product of $\vec{b}$ and $\vec{c}$ is -8.

Q: What is the final answer?

A: The final answer is $\boxed{-8}$.

Q: What is the significance of the dot product?

A: The dot product is a fundamental concept in mathematics and physics, and it has numerous applications in various fields. It is used to find the angle between two vectors, and it is also used in the calculation of work and energy.

Q: Can you provide more examples of the dot product?

A: Yes, here are a few more examples of the dot product:

• If $\vec{a} = \langle 2, 3 \rangle$ and $\vec{b} = \langle 4, 5 \rangle$, then $\vec{a} \cdot \vec{b} = 2 \cdot 4 + 3 \cdot 5 = 23$.
• If $\vec{a} = \langle 1, 2 \rangle$ and $\vec{b} = \langle 3, 4 \rangle$, then $\vec{a} \cdot \vec{b} = 1 \cdot 3 + 2 \cdot 4 = 11$.

I hope these examples help to illustrate the concept of the dot product. Let me know if you have any further questions.