Detailed Notes on Vector Operations for JEE

1. Vectors

• Definition: A vector is a mathematical entity with both magnitude and direction. It is represented by a directed line segment with an arrow indicating the direction.

• Representation: A vector is often represented using boldface notation, e.g., A. In component form, a vector can be expressed as A = (x, y, z), where x, y, and z are the components of the vector along the x, y, and z axes, respectively.

• Magnitude: The magnitude of a vector is the length of the directed line segment representing it. It is denoted by the symbol |**A**| or simply ||**A**||.

• Direction: The direction of a vector is the orientation of the directed line segment representing it. It is usually specified using angles measured with respect to a reference direction.

• Unit Vectors: A unit vector is a vector with magnitude 1. It is often used as a basis vector for representing other vectors. The standard unit vectors along the x, y, and z axes are denoted by i, j, and k, respectively.

2. Vector Addition and Subtraction

• Vector Addition: The vector sum of two vectors A and B is a new vector C that is obtained by placing the tail of B at the head of A and drawing a vector from the tail of A to the head of B. The vector addition operation is commutative, i.e., A + B = B + A. It is also associative, i.e., (A + B) + C = A + (B + C).

• Vector Subtraction: The vector difference of two vectors A and B is a new vector C that is obtained by reversing the direction of B and adding it to A. The vector subtraction operation is not commutative, i.e., A - B ≠ B - A. However, it is associative, i.e., (A - B) - C = A - (B - C).

• Properties of Vector Addition and Subtraction:

  • Commutative Property: A + B = B + A
  • Associative Property: (A + B) + C = A + (B + C)
  • Distributive Property: k(A + B) = kA + kB

• Triangle Law of Vector Addition: This law states that the vector sum of two vectors A and B is represented by the diagonal of the parallelogram formed by the two vectors.

Reference: NCERT Class 11 Physics textbook, Ch 4- Motion in a Plane: Vectors

• Parallelogram Law of Vector Addition: This law is a generalization of the triangle law and states that the vector sum of two vectors is represented by the diagonal of the parallelogram whose adjacent sides are parallel and equal in magnitude to the two vectors.

3. Scalar and Vector Products

• Scalar Product (Dot Product): The scalar product (or dot product) of two vectors A and B is a real number that is defined as the product of their magnitudes and the cosine of the angle between them. It is denoted by the symbol A • B. The scalar product is a commutative and distributive operation.

• Vector Product (Cross Product): The vector product (or cross product) of two vectors A and B is a new vector that is perpendicular to both A and B. It is denoted by the symbol A × B. The vector product is anti-commutative, i.e., A × B = -B × A, and distributive over vector addition.

• Properties of Scalar and Vector Products:

  • Commutative Property of Dot Product: A • B = B • A
  • Distributive Property of Dot Product: A • (B + C) = A • B + A • C
  • Anti-Commutative Property of Cross Product: A × B = -B × A
  • Distributive Property of Cross Product: A × (B + C) = A × B + A × C

• Geometrical Interpretations:

  • Dot Product: The dot product of two vectors represents the projection of one vector onto the other. It is a measure of the parallelism or anti-parallelism between two vectors.
  • Cross Product: The cross product of two vectors represents the area of the parallelogram formed by the two vectors. It is a measure of the perpendicularity between two vectors.

Reference: NCERT Class 12 Physics textbook, Ch 4- Moving Charges and Magnetism: Section 4.3

4. Scalar Triple Product

• The scalar triple product of three vectors A, B, and C is a real number defined as the determinant of the matrix formed by the components of the three vectors: (A • B) C − (A • C) B. It is denoted by the symbol [A B C].

• Properties of Scalar Triple Product:

• Cyclic Property: [A B C] + [B C A] + [C A B] = 0.

• Anti-commutative Property: [A B C] = -[B A C].

• Geometrical Interpretation: The scalar triple product represents the volume of the parallelepiped formed by the three vectors. It is a measure of the non-coplanarity of three vectors.

5. Vector Triple Product

• The vector triple product of three vectors A, B, and C is a new vector that is defined as the cross product of A and the cross product of B and C: A × (B × C). It is denoted by the symbol [A B C].

• Properties of Vector Triple Product:

  • Cyclic Property: [A B C] + [B C A] + [C A B] = 0.
  • Anti-commutative Property: [A B C] = -[B A C].
  • Distributive Property: A × (B + C) = A × B + A × C.

• Geometrical Interpretation: The vector triple product represents a vector that is perpendicular to the plane containing the three vectors. It is a measure of the non-coplanarity of three vectors.

Reference: NCERT Class 12 Physics textbook, Ch 4- Moving Charges and Magnetism: Section 4.4

6. Applications of Vectors in Physics

• Kinematics and Dynamics: Vectors are used to describe displacement, velocity, and acceleration in kinematics. They are also used to describe forces and momentum in dynamics.

• Electromagnetism: Vectors are used to describe electric and magnetic fields, as well as the forces that they exert on charged particles.

• Fluid Mechanics: Vectors are used to describe the velocity and acceleration of fluids, as well as the forces that they exert on objects.

• Thermodynamics: Vectors are used to describe the pressure, temperature, and entropy of thermodynamic systems.

7. Applications of Vectors in Engineering

• Structural Mechanics: Vectors are used to analyze the forces and stresses in structures, such as buildings, bridges, and machines.

• Fluid Mechanics: Vectors are used to analyze the flow of fluids in pipes and channels, as well as the forces that they exert on objects.

• Thermodynamics: Vectors are used to design and analyze thermodynamic systems, such as engines and heat pumps.

• Electrical Engineering: Vectors are used to analyze electric circuits and electromagnetic devices, such as motors and generators.

8. Vector Equations

• Vector Equations of Lines and Planes: Vector equations can be used to represent lines and planes in three dimensions. The vector equation of a line passing through a point P(x₁, y₁, z₁) with direction vector d = (d_x, d_y, d_z) is given by: r = P + td, where t is a real parameter.

• Vector Equations of Spheres, Cylinders, and Cones: Vector equations can also be used to represent spheres, cylinders, and cones in three dimensions:

  • Sphere:

  (x - x<sub>c</sub>)^2 + (y - y<sub>c</sub>)^2 + (z - z<sub>c</sub>)^2 = R^2
  

  • Cylinder:

    (x - x<sub>c</sub>)^2 + (y - y<sub>c</sub>)^2 = R^