Difference between Scalar (Dot Product) and Vector Product

Scalar Product (Dot Product) and Vector Product (Cross Product) are two fundamental operations used in vector mathematics. Here's an overview of each, along with some applications:

Scalar Product (Dot Product)

1. Definition: The scalar product of two vectors (usually denoted as A and B) is a scalar quantity found by multiplying the magnitude of one vector by the magnitude of the component of the other vector in the same direction.
2. Formula: A · B = |A| * |B| * cos(θ), where θ is the angle between the vectors.
3. Properties: Commutative, distributive over addition, and it results in a scalar.

Applications:

• Work Done: The dot product is used to calculate work done by a force when it acts on an object through a certain displacement.
• Projection: It's used to project one vector onto another, helping to find the component of one vector in the direction of another.
• Angle between Vectors: The dot product can determine the angle between two vectors based on their dot product value and magnitudes.

Vector Product (Cross Product)

1. Definition: The vector product of two vectors (A and B) is another vector, perpendicular to the plane formed by A and B. Its magnitude is the product of the magnitudes of A and B, multiplied by the sine of the angle θ between them.
2. Formula: A × B = |A| * |B| * sin(θ) * n, where n is the unit vector normal to the plane formed by A and B.
3. Properties: Non-commutative, distributive over addition, and it results in a vector.

Applications:

• Torque: In physics, the cross product is used to calculate the torque experienced by a body due to a force applied at a certain distance from an axis of rotation.
• Magnetic Fields: In electromagnetism, the cross product is used to find the magnetic field produced by a current-carrying wire.
• Angular Momentum: It's used to calculate the angular momentum of a rotating body.