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**(i)** internally**(ii)** externally

#### Solution

Let `bar("p"), bar("q")` and `bar("r")` be the position vectors of points P, Q and R respectively.

∴ `bar("p") = 2hat"i" - hat"j" + 3hat"k", bar("q") = -5hat"i" + 2hat"j" - 5hat"k"`, m:n = 3:2

**(i)** R divides the line PQ internally in the ratio 3:2

∴ By using section formula for internal division,

`bar("r") = ("m"bar("q") + "n"bar("p"))/("m" + "n")`

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

= `(-15hat"i" + 6hat"j" - 15hat"k" + 4hat"i" - 2hat"j" + 6hat"k")/5`

∴ `bar("r") = (-11hat"i" + 4hat"j" - 9hat"k")/5`

= `(-11)/5hat"i" + 4/5hat"j" - 9/5hat"k"`

**(ii)** R divides the line PQ externally in ratio 3:2

∴ By using section formula for external division,

`bar("r") = ("m"bar("q") - "n"bar("p"))/("m" - "n")`

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

= `(-15hat"i" + 6hat"j" - 15hat"k" - 4hat"i" + 2hat"j" - 6hat"k")/1`

∴ `bar("r") = -19hat"i" + 8hat"j" - 21hat"k"`