Basis Vectors

If you do not constrain i hat or j hat, then its possible to reach any point of the coordinate system. 

if both are 0, then its stuck in origin. 

 

Basis Vectors vs Normalized/Unit Vectors

Basis Vectors:

• Basis vectors are a set of vectors that form the basis of a vector space. They are often chosen to be mutually orthogonal (perpendicular) and linearly independent, meaning that no vector in the set can be expressed as a linear combination of the others. In 3D space, a common set of basis vectors is the standard unit vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$, each pointing along one of the coordinate axes (x, y, and z).
• Basis vectors are not required to have a unit length (magnitude of 1). They simply define the directions along which other vectors in the space can be expressed.

Normalized Vectors:

• A normalized vector is a vector that has been scaled to have a length (magnitude) of 1. It is obtained by dividing each component of the original vector by its magnitude. If $\mathbf{v}$ is a vector, its normalized counterpart is often denoted as �^v^.
• Normalized vectors are used when the direction of a vector is important, but its magnitude is not. Normalization simplifies calculations involving vectors and is particularly useful in algorithms, such as those related to computer graphics, physics simulations, or machine learning.

In summary, while basis vectors are a set of vectors chosen to form a basis for a vector space, normalized vectors are vectors that have been scaled to have a length of 1. Basis vectors may or may not be normalized, depending on the specific context or requirements of a problem.