Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Explore in-depth answers to your questions from a knowledgeable community of experts across different fields. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Sure, let's go through each of the statements one by one, providing a detailed explanation for each.
### Part I: True or False Statements
1. The dot product of two vectors is a vector.
- Explanation: The dot product of two vectors results in a scalar quantity, not a vector.
- Answer: False
2. Any vectors whose magnitude is zero is called unit vector.
- Explanation: A unit vector is defined as a vector with a magnitude (length) of 1. A vector with zero magnitude is simply the zero vector.
- Answer: False
3. If \( u = 2i + j \) and \( v = -2i - j \), then vector \( u \) and \( v \) are parallel vectors.
- Explanation: Two vectors are parallel if one is a scalar multiple of the other. In this case, \( v = -1 \cdot u \), making \( u \) and \( v \) parallel.
- Answer: True
4. The radius of the circle \( (x+2)^2 + y^2 - 9 = 0 \) is 9.
- Explanation: The given circle equation can be written as \( (x + 2)^2 + y^2 = 9 \). This conforms to the standard circle equation \( (x - h)^2 + (y - k)^2 = r^2 \), where \( r \) is the radius. Here, \( r^2 = 9 \) implies \( r = 3 \).
- Answer: False
5. If \( A \) is a square matrix that has two identical rows, then \(\operatorname{det}(A) = 0 \).
- Explanation: A square matrix with two identical rows has a determinant of zero by properties of determinants.
- Answer: True
6. Every square matrix is invertible.
- Explanation: A square matrix is invertible if and only if its determinant is non-zero. Not every square matrix meets this criterion.
- Answer: False
7. If the determinant of a square matrix is zero, then the matrix is singular.
- Explanation: By definition, a matrix is considered singular if its determinant is zero.
- Answer: True
8. If \( u = (3, 5) \) and \( v = (-2, 2) \), then \( u \cdot v = 6 \).
- Explanation: The dot product of \( u \) and \( v \) is calculated as \( 3 \cdot (-2) + 5 \cdot 2 = -6 + 10 = 4 \).
- Answer: False
9. The image of the point \( (2, -3) \) after being reflected about the line \( L: x = 0 \) is \( (-2, -3) \).
- Explanation: Reflecting a point \( (x, y) \) across the line \( x = 0 \) results in \( (-x, y) \). So, \( (2, -3) \) reflects to \( (-2, -3) \).
- Answer: True
10. Rigid motion is a motion that preserves distance.
- Explanation: By definition, rigid motion (or isometry) preserves distances between points.
- Answer: True
### Part II: Choose the Correct Answer from Given Alternatives
11. Which of the following is a scalar quantity?
- Explanation: Scalar quantities are those that are described by a magnitude (or numerical value) alone. Examples include mass, temperature, and speed, which do not depend on direction.
- Specific example: Mass
Thus, the complete and accurate answer summary is:
### Part I Summary
1. False
2. False
3. True
4. False
5. True
6. False
7. True
8. False
9. True
10. True
### Part II Summary
11. Mass
### Part I: True or False Statements
1. The dot product of two vectors is a vector.
- Explanation: The dot product of two vectors results in a scalar quantity, not a vector.
- Answer: False
2. Any vectors whose magnitude is zero is called unit vector.
- Explanation: A unit vector is defined as a vector with a magnitude (length) of 1. A vector with zero magnitude is simply the zero vector.
- Answer: False
3. If \( u = 2i + j \) and \( v = -2i - j \), then vector \( u \) and \( v \) are parallel vectors.
- Explanation: Two vectors are parallel if one is a scalar multiple of the other. In this case, \( v = -1 \cdot u \), making \( u \) and \( v \) parallel.
- Answer: True
4. The radius of the circle \( (x+2)^2 + y^2 - 9 = 0 \) is 9.
- Explanation: The given circle equation can be written as \( (x + 2)^2 + y^2 = 9 \). This conforms to the standard circle equation \( (x - h)^2 + (y - k)^2 = r^2 \), where \( r \) is the radius. Here, \( r^2 = 9 \) implies \( r = 3 \).
- Answer: False
5. If \( A \) is a square matrix that has two identical rows, then \(\operatorname{det}(A) = 0 \).
- Explanation: A square matrix with two identical rows has a determinant of zero by properties of determinants.
- Answer: True
6. Every square matrix is invertible.
- Explanation: A square matrix is invertible if and only if its determinant is non-zero. Not every square matrix meets this criterion.
- Answer: False
7. If the determinant of a square matrix is zero, then the matrix is singular.
- Explanation: By definition, a matrix is considered singular if its determinant is zero.
- Answer: True
8. If \( u = (3, 5) \) and \( v = (-2, 2) \), then \( u \cdot v = 6 \).
- Explanation: The dot product of \( u \) and \( v \) is calculated as \( 3 \cdot (-2) + 5 \cdot 2 = -6 + 10 = 4 \).
- Answer: False
9. The image of the point \( (2, -3) \) after being reflected about the line \( L: x = 0 \) is \( (-2, -3) \).
- Explanation: Reflecting a point \( (x, y) \) across the line \( x = 0 \) results in \( (-x, y) \). So, \( (2, -3) \) reflects to \( (-2, -3) \).
- Answer: True
10. Rigid motion is a motion that preserves distance.
- Explanation: By definition, rigid motion (or isometry) preserves distances between points.
- Answer: True
### Part II: Choose the Correct Answer from Given Alternatives
11. Which of the following is a scalar quantity?
- Explanation: Scalar quantities are those that are described by a magnitude (or numerical value) alone. Examples include mass, temperature, and speed, which do not depend on direction.
- Specific example: Mass
Thus, the complete and accurate answer summary is:
### Part I Summary
1. False
2. False
3. True
4. False
5. True
6. False
7. True
8. False
9. True
10. True
### Part II Summary
11. Mass
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
