What is position vector ? What is displacement vector ? Explain equality of vectors.

Position vector: To describe the position of an object moving in a plane, we need to choose a convenient point, say $\mathrm{O}$ as origin.

Let $\mathrm{P}$ and $\mathrm{P}^{\prime}$ be the positions of the object at time $t$ and $t^{\prime}$, respectively from figure (a). $\overrightarrow{O P}$ is the position vector of the object at time $t$. It is represented by a symbol $\vec{r}$.

Point $P^{\prime}$ is represented by another position vector. $\overrightarrow{O P^{\prime}}$ denoted by $\overrightarrow{r^{\prime}}$.

The length of the vector $\vec{r}$ represents the magnitude of the vector and its direction is the direction in which $P$ lies as seen from $O$.

Displacement vector : If the object moves from $\mathrm{P}$ to $\mathrm{P}^{\prime}$, the vector $\overrightarrow{P P}^{\prime}$ (with tail at $\mathrm{P}$ and tip at $\mathrm{P}^{\prime}$ ) is called the displacement vector corresponding to motion from point $\mathrm{P}$ (at time $t$ ) to point $\mathrm{P}^{\prime}$ (at time $t^{\prime}$ ).

Two vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ are said to be equal if and only if they have the same magnitude and the same direction.

Figure $(a)$ shows two equal vectors $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ can easily check their equality.

Shift $\overrightarrow{\mathrm{B}}$ parallel to itself until its tail $\mathrm{Q}$ coincides with that of $\mathrm{A}$, i.e. $\mathrm{Q}$ coincides with $\mathrm{O}$. Then, since their tips $\mathrm{S}$ and $\mathrm{P}$ also coincide. The two vectors are said to be equal.

Equality is indicated as $\overrightarrow{\mathrm{A}}=\overrightarrow{\mathrm{B}}$.

$(b)$

Two vectors $\vec{A}$ and $\vec{B}$ are said to be equal if and only if they have the same magnitude and the same direction.