Completeness of a set of basis vectors in 3D Euclidean space refers to the property that any vector in 3D space can be represented as a linear combination of the basis vectors. In other words, the basis vectors span the entire space and can be used to describe any point in 3D space.
Completeness of a set of basis vectors is important because it allows us to represent any vector in 3D space using a finite number of basis vectors. This makes it easier to perform calculations and manipulate vectors in 3D space.
A set of basis vectors is considered complete if it is linearly independent and spans the entire 3D space. This means that no basis vector can be expressed as a linear combination of the other basis vectors, and that every vector in 3D space can be represented as a linear combination of the basis vectors.
If a set of basis vectors is not complete, it means that there are certain vectors in 3D space that cannot be represented using the given basis. This can lead to inaccuracies and errors in calculations and may limit the range of vectors that can be described using the basis.
No, a set of basis vectors cannot be both complete and incomplete in 3D Euclidean space at the same time. It is either complete or incomplete based on the criteria mentioned above. However, a set of basis vectors can be considered complete in one 3D space and incomplete in another, depending on the dimensionality and properties of the space.