Math Insight

The zero vector

We define a vector as an object with a length and a direction. However, there is one important exception to vectors having a direction: the zero vector, i.e., the unique vector having zero length. With no length, the zero vector is not pointing in any particular direction, so it has an undefined direction.

We denote the zero vector with a boldface $\mathbf{0}$, or if we can't do boldface, with an arrow $\vec{0}$. It behaves essentially like the number 0. If we add $\vc{0}$ to any vector $\vc{a}$, we get the vector $\vc{a}$ back again unchanged.

For a given number of dimensions, there is only one vector of zero length (which justifies referring to this vector as the zero vector). We do, though, get a different zero vector depending on how many dimensions we are dealing with. In terms of components, the zero vector in two dimensions is $\vc{0} = (0,0)$, and the zero vector in three dimensions is $\vc{0}=(0,0,0)$. If we are feeling adventurous, we don't even need to stop with three dimensions. If we have an arbitrary number of dimensions, the zero vector is the vector where each component is zero.