Explain the null vector. Explain the physical significance of the null vector.

(N/A) Null vector: $A$ vector having zero magnitude and an indeterminate direction is called a null vector or zero vector. It is represented by $\vec{0}$.
Mathematical definition: If a vector $\vec{A}$ is subtracted from itself,the result is a null vector: $\vec{A} - \vec{A} = \vec{0}$.
Properties of a null vector:
$(i)$ $\vec{A} + \vec{0} = \vec{A}$
(ii) $\lambda \vec{0} = \vec{0}$ (where $\lambda$ is a scalar)
(iii) $0 \cdot \vec{A} = \vec{0}$
Physical Significance:
The null vector is essential to ensure that the result of vector operations remains a vector. For example,the displacement of a particle that returns to its starting point is a null vector.
As shown in the figure,a particle is at position $P$ at time $t=0$ with position vector $\vec{r}$. At time $t$,it moves to position $P'$ with position vector $\vec{r}'$. If the particle returns to $P$,the displacement $\Delta \vec{r} = \vec{r} - \vec{r} = \vec{0}$,which is a null vector.