Vector Multiplication (Dot Product and Cross Product) - SS1 Physics Lesson Note

Vector multiplication refers to two different operations performed on vectors: the dot product (also known as scalar product) and the cross product (also known as vector product). Let's discuss each operation:

1. Dot Product (Scalar Product):

The dot product of two vectors results in a scalar quantity. It calculates the scalar value that represents the magnitude of the projection of one vector onto the other. The dot product is denoted by a dot (·) or by the absence of a symbol between the vectors.

   The formula for the dot product between two vectors A and B is:

   A · B = |A| |B| cos(θ)

   where |A| and |B| represent the magnitudes of vectors A and B, and θ is the angle between them.

 

   Properties of the dot product:

   - The dot product of two parallel vectors equals the product of their magnitudes.

   - The dot product is commutative: A · B = B · A.

   - The dot product is distributive: A · (B + C) = A · B + A · C.

 The dot product is useful for determining the angle between two vectors, calculating work done, finding the projection of one vector onto another, and determining the magnitude of a vector.

 

2. Cross Product (Vector Product):

The cross product of two vectors results in a new vector that is perpendicular to both of the original vectors. The cross product is denoted by a cross (×) or by the multiplication symbol between the vectors.

 

   The formula for the cross product between two vectors A and B is:

   A × B = |A| |B| sin(θ) n

   where |A| and |B| represent the magnitudes of vectors A and B, θ is the angle between them, and n is a unit vector perpendicular to both A and B, determined using the right-hand rule.

   Properties of the cross-product:

   - The cross product is not commutative: A × B = -B × A.

   - The magnitude of the cross product is given by |A × B| = |A| |B| sin(θ).

   - The cross product is distributive: A × (B + C) = A × B + A × C.

The cross-product is useful in calculating torque, determining the direction of magnetic fields, finding the area of a parallelogram formed by two vectors, and solving problems involving rotational motion.

Vector multiplication operations, such as the dot product and cross product, play a fundamental role in vector analysis, mechanics, electromagnetism, and other branches of physics. They provide valuable insights into the relationships and interactions between vectors and are used in various mathematical calculations and physical applications.