What is Dot Product?
The Dot Product, also known as the Scalar Product, is an operation in mathematics that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In simpler terms, it multiplies corresponding components of two vectors and adds the products together.
Dot Product Definition
The dot product of two vectors, denoted by a ⋅ b, is defined in two ways:
Algebraically: The dot product is the sum of the products of the corresponding entries of the two sequences of numbers.
[Tex]a ⋅ b = ∑ (a_i * b_i)[/Tex]
Where:
- a and b are the vectors.
- i iterates over all dimensions (1 to n, where n is the number of dimensions).
- ai and bi represent the corresponding components of vectors a and b.
Geometrically: It is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them.
a ⋅ b = |a| |b| cos(θ)
Where:
- |a| and |b| are the magnitudes (lengths) of vectors a and b, and
- θ is the angle between them.
Dot Product
Dot Product, a fundamental operation in mathematics, is a unique way of combining two vectors that results in a scalar. This operation, often symbolized by a centered dot, is dependent on the length of both vectors and the angle between them.
Intuitively, the Dot Product tells us how much two vectors point in the same direction. It essentially measures the relative direction of two vectors. When the angle between the vectors is small, indicating they point in a similar direction, the dot product is large. Conversely, when the vectors are perpendicular, the dot product is zero.
In the following sections of this article, we will delve deeper into the concept of dot product, exploring its algebraic and geometric definitions, properties, and applications in various fields.
Table of Content
- What is Dot Product?
- Formula of Dot Product
- Angle Between Two Vectors Using Dot Product
- Projection of a Vector
- Working Rule to Find The Dot Product of Two Vectors
- Matrix Representation of Dot Product
- Dot Product of Unit Vectors
- Properties of Dot Product
- Applications of Dot Product
- Solved Examples on Dot Product
- Practice Problems on Dot Product