Which Of The Following Is The Dot Product Of $u = 4i + 6j - 5k$ And $v = -10i - 8j + 10k$?A. − 3 -3 − 3 B. 60 C. − 138 -138 − 138 D. 0

Introduction

In vector calculus, the dot product is a fundamental operation used to calculate the amount of "similarity" between two vectors. It is a scalar value that represents the amount of "alignment" between the two vectors. In this article, we will explore the dot product of two given vectors, $u = 4i + 6j - 5k$ and $v = -10i - 8j + 10k$. We will calculate the dot product and discuss its significance in vector calculus.

What is the Dot Product?

The dot product of two vectors, $u$ and $v$, is denoted by $u \cdot v$ and is calculated as follows:

$$u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$$

where $u_1$, $u_2$, and $u_3$ are the components of vector $u$, and $v_1$, $v_2$, and $v_3$ are the components of vector $v$.

Calculating the Dot Product

To calculate the dot product of $u = 4i + 6j - 5k$ and $v = -10i - 8j + 10k$, we will use the formula above.

$$u \cdot v = (4)(-10) + (6)(-8) + (-5)(10)$$

$$u \cdot v = -40 - 48 - 50$$

$$u \cdot v = -138$$

Significance of the Dot Product

The dot product has several significance in vector calculus. It is used to calculate the amount of "similarity" between two vectors, which is useful in various applications such as:

• Calculating the angle between two vectors
• Calculating the magnitude of a vector
• Calculating the projection of one vector onto another

Conclusion

In conclusion, the dot product of $u = 4i + 6j - 5k$ and $v = -10i - 8j + 10k$ is $-138$. The dot product is a fundamental operation in vector calculus that is used to calculate the amount of "similarity" between two vectors. It has several significance in various applications and is an essential concept to understand in vector calculus.

References

• [1] "Vector Calculus" by Michael Corral
• [2] "Calculus" by Michael Spivak

Frequently Asked Questions

• What is the dot product? The dot product is a scalar value that represents the amount of "similarity" between two vectors.
• How is the dot product calculated? The dot product is calculated using the formula: $u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$
• What is the significance of the dot product? The dot product is used to calculate the amount of "similarity" between two vectors, which is useful in various applications such as calculating the angle between two vectors, calculating the magnitude of a vector, and calculating the projection of one vector onto another.
  Understanding the Dot Product in Vector Calculus: A Q&A Article =============================================================

Introduction

In our previous article, we explored the dot product of two given vectors, $u = 4i + 6j - 5k$ and $v = -10i - 8j + 10k$. We calculated the dot product and discussed its significance in vector calculus. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information on the dot product.

Q&A Section

Q1: What is the dot product?

A: The dot product is a scalar value that represents the amount of "similarity" between two vectors. It is calculated using the formula: $u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$

Q2: How is the dot product calculated?

A: The dot product is calculated using the formula: $u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$. This formula involves multiplying the corresponding components of the two vectors and summing the results.

Q3: What is the significance of the dot product?

A: The dot product is used to calculate the amount of "similarity" between two vectors, which is useful in various applications such as calculating the angle between two vectors, calculating the magnitude of a vector, and calculating the projection of one vector onto another.

Q4: Can the dot product be negative?

A: Yes, the dot product can be negative. This occurs when the two vectors are not aligned in the same direction.

Q5: Can the dot product be zero?

A: Yes, the dot product can be zero. This occurs when the two vectors are orthogonal (perpendicular) to each other.

Q6: How is the dot product used in real-world applications?

A: The dot product is used in various real-world applications such as:

• Calculating the angle between two vectors
• Calculating the magnitude of a vector
• Calculating the projection of one vector onto another
• Calculating the work done by a force
• Calculating the energy of a system

Q7: Can the dot product be used to calculate the distance between two points?

A: No, the dot product cannot be used to calculate the distance between two points. However, it can be used to calculate the angle between two vectors, which can be used to calculate the distance between two points.

Q8: Can the dot product be used to calculate the area of a triangle?

A: No, the dot product cannot be used to calculate the area of a triangle. However, it can be used to calculate the angle between two vectors, which can be used to calculate the area of a triangle.

Q9: Can the dot product be used to calculate the volume of a pyramid?

A: No, the dot product cannot be used to calculate the volume of a pyramid. However, it can be used to calculate the angle between two vectors, which can be used to calculate the volume of a pyramid.

Q10: Can the dot product be used to calculate the surface area of a sphere?

A: No, the dot product cannot be used to calculate the surface area of a sphere. However, it can be used to calculate the angle between two vectors, which can be used to calculate the surface area of a sphere.

Conclusion

In conclusion, the dot product is a fundamental operation in vector calculus that is used to calculate the amount of "similarity" between two vectors. It has several significance in various applications and is an essential concept to understand in vector calculus. We hope that this Q&A article has provided additional information and clarified any doubts on the dot product.

References

• [1] "Vector Calculus" by Michael Corral
• [2] "Calculus" by Michael Spivak

Frequently Asked Questions

• What is the dot product? The dot product is a scalar value that represents the amount of "similarity" between two vectors.
• How is the dot product calculated? The dot product is calculated using the formula: $u \cdot v = u_1v_1 + u_2v_2 + u_3v_3$
• What is the significance of the dot product? The dot product is used to calculate the amount of "similarity" between two vectors, which is useful in various applications such as calculating the angle between two vectors, calculating the magnitude of a vector, and calculating the projection of one vector onto another.