because the formula for unit vector is full vector upon magnitude... here both the units cut off and only direction remains

Unit Vector: Unitless and Dimensionless

A unit vector is a vector that has a magnitude of 1. It is commonly used in mathematics and physics to represent directions or orientations. In contrast to regular vectors, which have both magnitude and direction, unit vectors are dimensionless and unitless. This can be explained in detail as follows:

Unit Vector Definition:
A unit vector is a vector that has a magnitude of 1. It is often denoted by a caret symbol (^) placed above the vector symbol. For example, if we have a vector A, its unit vector is represented as â.

Properties of Unit Vectors:
1. Magnitude: The magnitude of a unit vector is always equal to 1. It represents the length or size of the vector and is a scalar quantity. By definition, a unit vector's magnitude is always 1, regardless of the coordinate system or dimensions involved.

2. Direction: The direction of a unit vector is significant, representing the orientation of the vector. However, unit vectors do not have any units associated with their direction. They are dimensionless and unitless.

Unitless:
Unitless quantities are those that do not have any units associated with them. They are pure numbers and are not measured or expressed in terms of any particular physical unit. Unit vectors fall into this category because their magnitude is fixed at 1, regardless of the system of measurement or the physical quantities they represent.

Dimensionless:
Dimensionless quantities are those that do not have any physical dimensions. They are ratios or pure numbers obtained by dividing one physical quantity by another, canceling out the units. Since unit vectors do not have any units associated with their direction, they are considered dimensionless.

Reasoning:
The reason why unit vectors are unitless and dimensionless lies in their definition and properties. By definition, a unit vector has a magnitude of 1, which means it is not measured in any specific physical unit. Additionally, since the direction of a unit vector is independent of units, it is considered dimensionless. Therefore, unit vectors do not possess any physical dimensions or units, making them both unitless and dimensionless.

In conclusion, a unit vector is a vector with a magnitude of 1, representing directions or orientations. It is unitless and dimensionless because its magnitude is fixed at 1 and its direction does not have any associated physical dimensions.