AKTU Btech Unit 01 COORDINATE SYSTEMS covers numerous coordinate systems, transformations, vector operations, and their applications in geometry, physics, and engineering. Improve spatial representation and problem-solving abilities.
Dudes 🤔.. You want more useful details regarding this subject. Please keep in mind this as well. Important Questions For Electromagnetic Field Theory: *Unit-01 *Unit-02 *Unit-03 *Unit-04 *Unit-05 *Short-Q/Ans *Question-Paper with solution 21-22
Q1. What is a dot product ?.Also mention its properties and applications.
Ans. A. Scalar (or Dot) product:
1. The scalar or dot product of the two vectors A and B (you can take reference from image how to write vector A and B) is defined as the product of the magnitude of vector A, the magnitude of vector B and the cosine of the smaller angle between them.
![What is a dot product ?.Also mention its properties and applications](https://bachelorexam.com/wp-content/uploads/2023/05/image-421.png)
![What is a dot product ?.Also mention its properties and applications](https://bachelorexam.com/wp-content/uploads/2023/05/image-421.png)
B. Properties of dot product:
1. If the two vectors are parallel to each other, i.e. 𝝷 = 0° then cos 𝝷AB =1
![What is a dot product ?.Also mention its properties and applications](https://bachelorexam.com/wp-content/uploads/2023/05/image-424.png)
![What is a dot product ?.Also mention its properties and applications](https://bachelorexam.com/wp-content/uploads/2023/05/image-424.png)
C. Applications of dot product:
1. To find the angle between the two vectors.
![What is a dot product ?.Also mention its properties and applications](https://bachelorexam.com/wp-content/uploads/2023/05/image-422.png)
![What is a dot product ?.Also mention its properties and applications](https://bachelorexam.com/wp-content/uploads/2023/05/image-422.png)
3. Physically, work done by a constant force Vector F over a straight displacement Vector d can be expressed as a dot product of two vectors.
![What is a dot product ?.Also mention its properties and applications](https://bachelorexam.com/wp-content/uploads/2023/05/image-423.png)
![What is a dot product ?.Also mention its properties and applications](https://bachelorexam.com/wp-content/uploads/2023/05/image-423.png)
Q2. What is a cross product ? Also mention its properties.
Ans. A. Cross (or vector) product:
1. The cross or vector product is defined as the product of the magnitudes of vectors A and B (you can take reference from image how to write vector A and B) and the sine of the smaller angle between vectors A and B But this product is a vector quantity and has a direction perpendicular to the plane containing the two vector vectors A and B .
![What is a cross product ? Also mention its properties](https://bachelorexam.com/wp-content/uploads/2023/05/image-428.png)
![What is a cross product ? Also mention its properties](https://bachelorexam.com/wp-content/uploads/2023/05/image-428.png)
B. Properties of cross product:
1. Two vectors are considered parallel if their cross product is zero.
![What is a cross product ? Also mention its properties](https://bachelorexam.com/wp-content/uploads/2023/05/image-425.png)
![What is a cross product ? Also mention its properties](https://bachelorexam.com/wp-content/uploads/2023/05/image-425.png)
3. Cross product in determinant form: Consider the two vector in the cartesian system as,
![What is a cross product ? Also mention its properties](https://bachelorexam.com/wp-content/uploads/2023/05/image-426.png)
![What is a cross product ? Also mention its properties](https://bachelorexam.com/wp-content/uploads/2023/05/image-426.png)
![What is a cross product ? Also mention its properties](https://bachelorexam.com/wp-content/uploads/2023/05/image-427.png)
![What is a cross product ? Also mention its properties](https://bachelorexam.com/wp-content/uploads/2023/05/image-427.png)
Q3. Write a short note on cartesian coordinate system.
Ans. 1. It is also known as rectangular coordinate system.
2. In rectangular coordinate system, three coordinate axis, i.e., x, y and z are set up mutually at right angles to each other.
3. A point Pin cartesian coordinate system is represented by P(x,y,z).
4 The ranges of the coordinate variables x, y and z are
![Write a short note on cartesian coordinate system](https://bachelorexam.com/wp-content/uploads/2023/05/image-429.png)
![Write a short note on cartesian coordinate system](https://bachelorexam.com/wp-content/uploads/2023/05/image-429.png)
Q4. Give a brief description on cylindrical coordinate system ?
Ans. A. Cylindrical coordinate system:
1. A point P (ρ, ф, z) in cylindrical coordinate system represents p the radius of the cylinder, ф the azimuthal angle and z is same as in cartesian system.
2. The ranges of variables ρ, ф and z are
![Give a brief description on cylindrical coordinate system](https://bachelorexam.com/wp-content/uploads/2023/05/image-430.png)
![Give a brief description on cylindrical coordinate system](https://bachelorexam.com/wp-content/uploads/2023/05/image-430.png)
B. Relation between cartesian and cylindrical ocoordinate system:
![Give a brief description on cylindrical coordinate system](https://bachelorexam.com/wp-content/uploads/2023/05/image-431.png)
![Give a brief description on cylindrical coordinate system](https://bachelorexam.com/wp-content/uploads/2023/05/image-431.png)
Q5. Convert a point P(4, -3, 6) and a vector R = z ax + y az into cylindrical coordinate systems.
Ans. 1. At point P, x = 4, y =-3, 2 = 6
![Convert a point P(4, -3, 6) and a vector R = z ax + y az into cylindrical coordinate systems](https://bachelorexam.com/wp-content/uploads/2023/05/image-433.png)
![Convert a point P(4, -3, 6) and a vector R = z ax + y az into cylindrical coordinate systems](https://bachelorexam.com/wp-content/uploads/2023/05/image-433.png)
![Convert a point P(4, -3, 6) and a vector R = z ax + y az into cylindrical coordinate systems](https://bachelorexam.com/wp-content/uploads/2023/05/image-432.png)
![Convert a point P(4, -3, 6) and a vector R = z ax + y az into cylindrical coordinate systems](https://bachelorexam.com/wp-content/uploads/2023/05/image-432.png)
![Convert a point P(4, -3, 6) and a vector R = z ax + y az into cylindrical coordinate systems](https://bachelorexam.com/wp-content/uploads/2023/05/image-434.png)
![Convert a point P(4, -3, 6) and a vector R = z ax + y az into cylindrical coordinate systems](https://bachelorexam.com/wp-content/uploads/2023/05/image-434.png)
Q6. Describe the gradient of a scalar field.
Ans. A. Gradient of scalar field:
1. The gradient of a scalar field M is a vector that depicts the maximum rate of increase of M in space in terms of both magnitude and direction.
2. The operation of the ▽(del) operator on a scalar function is called gradient of a scalar
![Describe the gradient of a scalar field.](https://bachelorexam.com/wp-content/uploads/2023/05/image-435.png)
![Describe the gradient of a scalar field.](https://bachelorexam.com/wp-content/uploads/2023/05/image-435.png)
B. Properties of gradient of scalar:
- 1. The gradient ▽M gives the maximum rate of change of M per unit distance.
- 2. The gradient ▽M always indicates the direction of the maximum rate of change of M.
- 3.The gradient ▽M at any point is perpendicular to the constant M surface, which passes through the point.
- 4. The directional derivative of M along the unit vector a is ▽M. a , which is projection of ▽M in the direction of unit vector a.
![bachelor exam preparation all question paper with solution important questions with solution](http://bachelorexam.com/wp-content/uploads/2023/01/cute-teddy-4.png)
![bachelor exam preparation all question paper with solution important questions with solution](http://bachelorexam.com/wp-content/uploads/2023/01/cute-teddy-4.png)
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