Dot product
Definition
The dot product, also known as the scalar product, is a binary operation that takes two vectors and returns a scalar. It is denoted by a dot (·), hence the name "dot product". In the context of Euclidean geometry, the dot product of two vectors is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. This definition is often referred to as the geometric definition of the dot product vector.
Mathematical Representation
Given two vectors, A and B, each with dimensions n, the dot product can be mathematically represented as follows:
A · B = Σ (Ai * Bi) for i = 1 to n
Where Ai and Bi are the components of vectors A and B respectively. This definition is often referred to as the algebraic definition of the dot product.
Properties
The dot product has several important properties, including:
1. Commutativity: A · B = B · A 2. Distributivity over addition: A · (B + C) = A · B + A · C 3. Scalar multiplication: (kA) · B = k(A · B) where k is a scalar 4. Orthogonality: If A · B = 0, then A and B are orthogonal (or perpendicular)
These properties make the dot product a fundamental tool in the fields of linear algebra, physics, and computer science.
Applications
The dot product is used in a variety of scientific and mathematical fields. Some of its applications include:
1. Physics: In physics, the dot product is used in the definition of work done by a force. Work is defined as the dot product of the force vector and the displacement vector. 2. Computer Science: In computer graphics, the dot product is used to determine the angle between two vectors, which can be used to calculate lighting and shading in 3D models. 3. Machine Learning: In machine learning, the dot product is used in the calculation of the weighted sum of inputs in artificial neural networks.
See Also
