JEE Main & Advanced Mathematics Vector Algebra Position Vector

If a point \[O\] is fixed as the origin in space (or plane) and \[P\] is any point, then \[\overrightarrow{OP}\] is called the position vector of \[P\] with respect to \[O\].

If we say that P is the point \[\mathbf{r}\], then we mean that the position vector of \[P\] is \[\mathbf{r}\] with respect to some origin \[O\].

(1) \[\overrightarrow{AB}\] in terms of the position vectors of points A and B : If \[\mathbf{a}\] and b are position vectors of points A and B respectively. Then, \[\overrightarrow{OA}=\mathbf{a},\,\overrightarrow{OB}=\mathbf{b}\]

\[\therefore \] \[\overrightarrow{AB}\] =  (Position vector of B) – (Position vector of A)

\[=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf{b}-\mathbf{a}\]

(2) Position vector of a dividing point : The position vectors of the points dividing the line \[AB\]in the ratio m : n internally or externally are \[\frac{m\mathbf{b}+n\mathbf{a}}{m+n}\] or \[\frac{m\mathbf{b}-n\mathbf{a}}{m-n}\].