Full Answer Section

Vector and Scalar Quantities

a) Definition and Examples:

Scalar Quantity: A scalar quantity has only magnitude (size) and no direction. It can be represented by a single number.

• Examples: Temperature (25°C), mass (5 kg), speed (10 m/s), time (2 hours), distance (5 km), energy (100 Joules).

Vector Quantity: A vector quantity has both magnitude and direction. It can be represented by an arrow, where the length of the arrow represents the magnitude and the arrowhead indicates the direction.

• Examples: Displacement (5 km East), velocity (10 m/s North), force (10 N downward), acceleration (5 m/s² West), momentum (20 kg m/s upward).

Types of Vector Quantities:

• Displacement: The change in position of an object from a starting point to an endpoint.

• Velocity: The rate of change of displacement with time.

• Acceleration: The rate of change of velocity with time.

• Force: A push or pull on an object.

• Momentum: The product of an object’s mass and velocity.

b) Law of Parallelogram and Triangle of Vector Addition:

Law of Parallelogram: To add two vectors (A and B) using the parallelogram method, follow these steps:

1. Draw: Place the vectors tail-to-tail.

2. Complete: Complete the parallelogram by drawing parallel lines.

3. Diagonal: The diagonal of the parallelogram starting from the common tail represents the resultant vector (R).

Law of Triangle: To add two vectors (A and B) using the triangle method, follow these steps:

1. Draw: Place the vectors head-to-tail.

2. Complete: Draw a line connecting the tail of the first vector (A) to the head of the second vector (B).

3. Resultant: The resultant vector (R) starts at the tail of the first vector and ends at the head of the second vector.

c) Scalar (Dot) Product and Vector (Cross) Product:

Scalar Product (Dot Product):

• Definition: The dot product of two vectors is a scalar quantity equal to the product of their magnitudes and the cosine of the angle between them.

• Formula: A ⋅ B = |A| |B| cos θ

• Example: A force of 10 N is applied at an angle of 30° to a surface. The work done by the force is the dot product of the force and the displacement.

• Properties:

  • Commutative: A ⋅ B = B ⋅ A

  • Distributive: A ⋅ (B + C) = A ⋅ B + A ⋅ C

  • Zero Vector: A ⋅ 0 = 0

  • Orthogonal Vectors: If A and B are orthogonal (perpendicular), then A ⋅ B = 0.

Vector Product (Cross Product):

• Definition: The cross product of two vectors is a vector quantity whose magnitude is equal to the product of their magnitudes and the sine of the angle between them. The direction of the cross product is perpendicular to both vectors and is determined by the right-hand rule.

• Formula: A × B = |A| |B| sin θ n

• Example: The torque on a rotating object is the cross product of the force applied and the distance from the axis of rotation.

• Properties:

  • Anti-commutative: A × B = – (B × A)

  • Distributive: A × (B + C) = A × B + A × C

  • Zero Vector: A × 0 = 0

  • Parallel Vectors: If A and B are parallel, then A × B = 0.

The dot product is used for calculating work, energy, and power. The cross product is used for calculating torque, angular momentum, and magnetic force.