The Geometric Relationship Between Euclidean Distance, Euclidean Norm, and Scalar Product

Euclidean distance, Euclidean norm, and scalar product (dot product) are fundamental concepts in Euclidean geometry and linear algebra. These concepts are closely related to each other and provide a comprehensive framework for understanding geometric relationships in Euclidean space. In this article, we will explore the definitions of these concepts and delve into the geometric relationships between them.

1. Euclidean Distance

The Euclidean distance between two points (mathbf{a} (a_1, a_2, ldots, a_n)) and (mathbf{b} (b_1, b_2, ldots, b_n)) in n-dimensional space is defined as:

Euclidean Distance: [d(mathbf{a}, mathbf{b}) sqrt{sum_{i1}^{n} (a_i - b_i)^2}]

This distance measures the straight-line length between the two points.

2. Euclidean Norm

The Euclidean norm or length of a vector (mathbf{v} (v_1, v_2, ldots, v_n)) is defined as:

Euclidean Norm: [|mathbf{v}| sqrt{sum_{i1}^{n} v_i^2}]

This norm represents the length of the vector from the origin to the point defined by the vector in Euclidean space.

3. Scalar Product (Dot Product)

The scalar product or dot product of two vectors (mathbf{u} (u_1, u_2, ldots, u_n)) and (mathbf{v} (v_1, v_2, ldots, v_n)) is defined as:

Scalar Product (Dot Product): [mathbf{u} cdot mathbf{v} sum_{i1}^{n} u_i v_i]

This product provides a measure of how much one vector extends in the direction of another and can also be related to the angle between the two vectors.

Geometric Relationships

Distance and Norm

Distance and Norm: The distance between two points can be expressed in terms of the norms of their difference:

Distance Formula: [d(mathbf{a}, mathbf{b}) |mathbf{a} - mathbf{b}|]

This shows that the distance is simply the norm of the vector that points from (mathbf{b}) to (mathbf{a}).

Norm and Scalar Product

Norm and Scalar Product: The square of the norm of a vector can be expressed using the scalar product:

Norm Formula: [|mathbf{v}|^2 mathbf{v} cdot mathbf{v}]

This means that the length of a vector can be calculated as the square root of its dot product with itself.

Dot Product and Angle

Dot Product and Angle: The dot product can also be related to the angle (theta) between two vectors:

Dot Product Formula: [mathbf{u} cdot mathbf{v} |mathbf{u}| |mathbf{v}| cos theta]

This relationship shows how the scalar product incorporates both the magnitudes of the vectors and the cosine of the angle between them. This provides a way to calculate the angle if the vectors' lengths and their dot product are known.

Summary

In summary, the Euclidean distance is derived from the norm of the difference between two points. The norm itself is related to the scalar product of a vector with itself. The scalar product connects to the angle between vectors. Together, they form a coherent framework for understanding geometric relationships in Euclidean space.