# Position Vector

## Position Vector

Position vector is a vector which represents the position of a point in a space with respect to the origin. It also represents the distance and direction of the points from the origin. If A is the point then the position vector of A is represented by

O is the origin and A is any point in the space is the position vector, which represents the distance as well as the direction of the point from the origin.

Position vector is used to specify the position of a certain body. Knowing the position of a body is essential when it comes to describe the motion of that body.

Let, A be a point in space whose coordinates are (x, y, z), then we know that its position vector with respect to the origin of coordinates system is given by:

Position Vector $$\left( \overrightarrow{r} \right)\,\,=\,\,x\widehat{i}+y\widehat{j}+z\widehat{k}$$.

Where,

$$\widehat{i}$$= Unit vector along x direction,

$$\widehat{j}$$ = Unit vector along y direction,

$$\widehat{k}$$ = Unit vector along z direction.

How to find the Position Vector?

The position vector of point A (x₁, y₁, z₁) is given by:

$$\overrightarrow{OA}\,\,=\,\,\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)$$.

For Example: The position vector of a point A (2, -3, 5) is written as:

$$\overrightarrow{OA}\,\,=\,\,\left( 2,-3,5 \right)$$,

To find the position vector of a line joining point A (x₁, y₁, z₁) and B (x₂, y₂, z₂) is given by:

$$\overrightarrow{AB}\,\,=\,\,\left( \left( {{x}_{2}}-{{x}_{1}} \right),\left( {{y}_{2}}-{{y}_{1}} \right),\left( {{z}_{2}}-{{z}_{1}} \right) \right)$$.

For Example: To find the position vector of line joining A (4, -1, 3) and B (4, 5, -1 is written as:

$$\overrightarrow{AB}\,\,=\,\,\left( \left( 4-4 \right),\left( 5-\left( -1 \right) \right),\left( -1-3 \right) \right)=\left( 0,6,-4 \right)$$.