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A zero vector is a vector with zero length and no specific direction. Zero vector can also be defined as a vector whose initial and terminal points are at the same location. Graphically, a zero vector is represented by a single point.
Zero vector is denoted by or the number with an arrow above it.
Zero vector has several special properties that distinguish it from other vectors:
Zero vector is the identity element in vector addition operations. If is any vector, then
The magnitude (length) of zero vector is zero:
Zero vector cannot be normalized, as it is impossible to create a unit vector from zero vector.
Zero vector is unique in a vector space.
Zero vector is simultaneously parallel and perpendicular (orthogonal) to all other vectors.
The scalar product of zero vector with any real number will still result in zero vector:
In Cartesian coordinate form, zero vector can be expressed as:
One example of a zero vector in daily life is the displacement of someone who returns to their starting position. For instance:
If Andi walks east and then west, his total displacement is zero. Although Andi has traveled a total distance of , his displacement (the difference between final and initial positions) is zero.
If is any vector, then:
This shows that zero vector acts as the identity element in vector addition operations.
For any vector :
For any scalar :
The concept of zero vector is very important in solving various vector problems. Here are some applications:
