Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2hat"i" - hat"j" + 3hat"k"` and `- 5hat"i" + 2hat"j" - 5hat"k"` in the ratio 3 : 2 is externally.

#### Solution

It is given that the points P and Q have position vectors `bar"p" = 2hat"i" - hat"j" + 3hat"k"` and `bar"q" = - 5hat"i" + 2hat"j" - 5hat"k"` respectively.

If R(`bar"r"`) divides the line segment joining P and Q externally in the ratio 3 : 2, by section formula for external division,

`bar"r" = (3bar"q" - 2bar"p")/(3 - 2)`

`= (3(- 5hat"i" + 2hat"j" - 5hat"k") - 2(2hat"i" - hat"j" + 3hat"k"))/(3 - 2)`

`= -19hat"i" + 8hat"j" - 21hat"k"`

∴ coordinates of R = (- 19, 8, -21).

Hence, the position vector of R is `-19hat"i" + 8hat"j" - 21hat"k"` and coordinates of R are (-19, 8, -21).