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In a vector space with a bilinear form, a vector that is self-orthogonal (i.e. on which the bilinear form is zero) is referred to as a null vector. In a seminormed vector space, it refers to a vector with zero seminorm. In contrast, the term zero vector refers to the unique additive identity of the vector space.
In contexts in which the only null vector is the zero vector (such as Euclidean vector space) or where there is no defined concept of magnitude, null vector may be used as a synonym for zero vector.
The zero vector is unique: if a and b are zero vectors, then a = a + b = b.
The zero vector is a special case of the zero tensor. It is the result of scalar multiplication by the scalar 0 (here meaning the additive identity of the underlying field, not necessarily the real number 0).
The zero vector is, by itself, linearly dependent, and so any set of vectors which includes it is also linearly dependent.
The zero vector is both parallel and perpendicular to every vector.
- In a normed vector space there is only one vector of norm equal to 0. This is just the zero vector.
- In a seminormed vector space, there might be more than one vector of norm equal to 0. These vectors are often called null vectors.
- In an anisotropic vector space (e.g. Euclidean vector space), the zero vector is the only null vector.
- In a symplectic vector space, every vector is a null vector.
- A zero vector space is a vector space whose only element is the zero vector.
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