2: Aerodynamic Fundamental principles and equations

Question 1

What is a vector and how is it represented?

Answer 1

A vector is a quantity with both magnitude and direction. It is typically represented as an arrow, with the length representing its magnitude and the direction of the arrow showing its direction.

Question 2

What is the magnitude of a vector
A, and how is it denoted?

Answer 2

A2: The magnitude of a vector
A is the length of the vector and is denoted by
∣A∣. It is a scalar quantity.

Question 3

What is a unit vector?

Answer 3

A unit vector is a vector with a magnitude of 1, and it points in the same direction as the original vector. It is defined as n= A/∣A∣

Question 4

How do you add two vectors A and B?

Answer 4

The vector sum of A and B is given by A+B=C, where C is formed by connecting the tail of A to the head of B.

Question 5

How do you subtract one vector
B from another vector A?

Answer 5

The vector difference A−B is formed by connecting the tail of A to the head of −B, resulting in vector D.

Question 6

What is the formula for the dot product (scalar product) of two vectors A and B?

Answer 6

The scalar (dot) product is given by: A⋅B=∣A∣∣B∣cosθ
where θ is the angle between the two vectors.

Question 7

What does the scalar product represent geometrically?

Answer 7

The scalar product represents the magnitude of
A multiplied by the magnitude of the component of B along the direction of A.

Question 8

What is the formula for the cross product (vector product) of two vectors A and B?

Answer 8

The vector (cross) product is given by:
A×B=(∣A∣∣B∣sinθ)

Question 9

What is the right-hand rule, and how is it used in the cross product?

Answer 9

The right-hand rule states that when you curl the fingers of your right hand in the direction from
A to B, your thumb points in the direction of the resulting vector A×B.

Question 10

What are Cartesian coordinates, and how are they represented?

Answer 10

Cartesian coordinates are a system where the space is divided into three mutually perpendicular axes:
x, y, and z. The unit vectors along these axes are i, j, and k, respectively.

Question 11

How is a vector A represented in Cartesian coordinates?

Answer 11

A vector A in Cartesian coordinates is expressed as: A = AxI+Ayj+AzK

Where Ax, Ay, and Az are the scalar components of A along the x, y, and z axes.

Question 12

What are cylindrical coordinates, and how are they defined?

Answer 12

Cylindrical coordinates are defined by three variables:

r: radial distance from the origin in the xy-plane,

θ: angle from the positive x-axis in the xy-plane,

z: height along the z-axis.

Question 13

How is a vector A expressed in cylindrical coordinates?

Answer 13

A vector A in cylindrical coordinates is expressed as: A = Area + Athetaetheta + Azez

Where Ar, Atheta, and Az are components along the r, theta, and z directions respectively.

Question 14

How do you transform between Cartesian and cylindrical coordinates?

Answer 14

The transformation equations are:

x=rcosθ

y=rsinθ

z=z

Question 15

What are spherical coordinates, and how are they defined?

Answer 15

Spherical coordinates are defined by three variables:

r: distance from the origin,

θ: angle from the z-axis,

ϕ: angle in the xy-plane from the x-axis.

Question 16

How is a vector A expressed in spherical coordinates?

Answer 16

A vector A in spherical coordinates is expressed as: A = Arer + Athetaetheta + Aphiephi

Where Ar, Atheta, and Aphi are components along the r, theta, and phi directions respectively.

Question 17

What is a scalar field?

Answer 17

A scalar field is a scalar quantity that varies with position and time, e.g., pressure p(x,y,z,t), density ρ(x,y,z,t), and temperature T(x,y,z,t).

Question 18

How do you transform between Cartesian and spherical coordinates?

Answer 18

The transformation equations are:

x=rsinθcosϕ

y=rsinθsinϕ

z=rcosθ

Question 19

What is a vector field?

Answer 19

A vector field is a vector quantity that varies with position and time, e.g., velocity V(x,y,z,t)=Vxi + Vyj + Vzk

Question 20

How do you compute the dot product of two vectors
A and B in Cartesian coordinates?

Answer 20

The dot product is calculated as:

A⋅B=AxBx + AyBy + AzBz

Question 21

How do you compute the cross product of two vectors A and B in Cartesian coordinates?

Answer 21

The cross product is calculated using the determinant

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