Zero Vector

A vector whose magnitude is zero is called a zero vector or null vector. So, $(0,0)$ is the zero vector. In a zero vector, the origin and the terminal point coincide. The direction of a zero vector is indeterminate.

A zero vector is represented by $\overrightarrow{O}$. A vector when multiplied by $0$ gives a zero vector. \[\therefore\overrightarrow{O}=\overrightarrow{0}\]

Properties of a Zero Vector

  1. The addition of a zero vector to a given vector or subtraction of the zero vector from the given vector does not alter the given vector. Thus, \[\overrightarrow{A}+\overrightarrow{O}=\overrightarrow{A}\] \[\overrightarrow{A}-\overrightarrow{O}=\overrightarrow{A}\]
  2. If a zero vector is multiplied by a non zero scalar $n$, then again a zero vector is obtained. \[\therefore n\overrightarrow{O}=\overrightarrow{O}\]
  3. If $m$ and $n$ are two different non zero scalars, then the relation \[m\overrightarrow{A}=n\overrightarrow{B}\] can hold only if both $\overrightarrow{A}$ and $\overrightarrow{B}$ are zero vectors.

A zero vector has a lot of physical significance. It is useful in describing the physical situation involving vector quantities.

Examples of Zero Vectors

  1. Velocity vector of a stationary particle is a zero vector.
  2. The acceleration vector of an object moving with a uniform velocity is a zero vector.
  3. The position vector of the origin of coordinate axes is a zero vector.


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