Unit 01 COORDINATE SYSTEMS Important Question Answer AKTU Btech

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.

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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.

B. Properties of dot product: 

1. If the two vectors are parallel to each other, i.e. 𝝷 = 0° then cos 𝝷_AB =1 

C. Applications of dot product: 

1. To find the angle between the two vectors.

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.

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 .

B. Properties of cross product: 

1. Two vectors are considered parallel if their cross product is zero.  

3. Cross product in determinant form: Consider the two vector in the cartesian system as,

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

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

B. Relation between cartesian and cylindrical ocoordinate system: 

Q5. Convert a point P(4, -3, 6) and a vector R = z a_x + y a_z into cylindrical coordinate systems.

Ans. 1. At point P, x = 4, y =-3, 2 = 6

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  

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.

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