Position vector in two-dimensional cartesian co-ordinate system:

A particle is moving in a plane so we will use a two-dimensional co-ordinate system in this case to describe the motion of the particle.

Let us consider the motion in X-Y plane.

Let at any instant of time \( x \) and \( y \) be the Cartesian co-ordinates of the particle P.

The position vector of the particle  \( P \) is  \( \vec{OP}=\vec{r} \).

We can write, \( \vec{OP}=\vec{OQ}+\vec{QP} \)

\hat{i} and \hat{j} are the unit vectors along X-axis and Y-axis respectively

So, \( \vec{OQ}=x\hat{i} \) and \( \vec{QP}=y\hat{j} \)

Therefore,  \( \vec{r}=x\hat{i}+y\hat{j} \)

\( \vec{r}\cdot\vec{r}=[ x\hat{i}+y\hat{j}]\cdot[x\hat{i}+y\hat{j}] \)

Or, \( r^2=x^2+y^2 \)

Or, \( |\vec{r}|=\sqrt{x^2+y^2} \)

Where, \( |\vec{r}| \) is the magnitude of the position vector  \( \vec{r} \).

Position vector in the three-dimensional Cartesian co-ordinate system:

Let us consider a particle P is in the space and x, y, and z be the three Cartesian co-ordinates in the three-dimensional Cartesian co-ordinate system. OX, OY, and OZ are the three mutually perpendicular axes meeting at the origin O.

The position vector of the particle P with respect to the origin O is  \( \vec{r} \).

OX, OY, and OZ represent the X, Y, and Z axes respectively.

\( \hat{i} \),  \( \hat{j} \) and  \( \hat{k} \) are the unit vectors along the X, Y, and Z axes respectively.

Draw a perpendicular line PQ from the point P on the X-Y plane.

Draw a line QR parallel to the Y-axis, meeting the X-axis at the point R. Draw another line QS parallel to the X-axis, meeting the Y-axis at the point S.

Here, \( \vec{OR}=x\hat{i} \), \( \vec{OS}=y\hat{j} \), and \( \vec{QP}=z\hat{k} \).

Again \( \vec{OQ}=\vec{OR}+\vec{OS} \)

Or, \( \vec{OQ}=x\hat{i}+y\hat{j} \)

\( \vec{OP}=\vec{OQ}+\vec{QP}=\vec{OR}+\vec{OS}+\vec{QP} \)

[since,  \( \vec{OQ}=\vec{OR}+\vec{OS} \) ]

Or, \( \vec{OP}=x\hat{i}+y\hat{j}+z\hat{k} \)

Or, \( \vec{r}= x\hat{i}+y\hat{j}+z\hat{k} \)

Now, \( \vec{r}\cdot\vec{r}=( x\hat{i}+y\hat{j}+z\hat{k})\cdot(x\hat{i}+y\hat{j}+z\hat{k}) \)

Or, \( r^2=x^2+y^2+z^2 \)

Or, \( |\vec{r}|=\sqrt{(x^2+y^2+z^2)} \).

Where, \( |\vec{r}| \) is the magnitude of the vector \( \vec{r} \)

\( \hat{r} \) is the unit vector along the vector \( \vec{r} \)

Then, \( \displaystyle{\hat{r}=\frac{\vec{r}}{|\vec{r}|}} \)