Final answer:
The zero vector, also known as the null vector, is the vector in any vector space that has no length and no direction, with all its components being zero. It serves as the identity element for vector addition and has the property that it is orthogonal to all other vectors.
Step-by-step explanation:
The zero vector of a vector space is a special element that acts as an identity with respect to vector addition. It is called the null vector and is denoted by 0. Equivalent to the number zero in scalar algebra, the null vector has all its components equal to zero, meaning it is represented as 0 = 0î + 0ĸ + 0k in a three-dimensional Cartesian coordinate system. Consequently, this vector has no length and no direction, and when added to any vector, it leaves that vector unchanged.
In the context of vector operations, the zero vector is unique in that it does not affect other vectors during addition. For instance, if à is a vector in this space, adding it to the zero vector results in the same vector Ã, emphasizing the commutative property of vector addition (à + 0 = 0 + à = Ã). Additionally, the zero vector is orthogonal to all other vectors, as seen in scalar products where the dot product of any vector with the null vector results in zero (à ⋅ 0 = 0).