The position vector of point R dividing the line segment joining two points P and Q in the ratio m: n is given by:i. Internally:m+nmb+na​ ii. Externally:m−nmb−na​ Position vectors of Pand Q are given as:OP=i^+2j^​−k^ and OQ​=−i^+j^​+k^ Position vectors of P and Q are given as:OP=i^+2j^​−k^ and OQ​=−i^+j^​+k^ (i) The position vector of point R which divides the line joining two points P and Q internally in the ratio 2: 1 is given by;OR​=2+12(−i^+j^​+k^)+1(i^+2j^​−k^)​=3(−2i^+2j^​+2k^)+(i^+2j^​−k^)​=3−i^+4j^​+k^​=−31​i^+34​j^​+31​k^​ (ii) The position vector of point R which divides the line joining two points P and Q externally in the ratio 2: 1 is given by,OR​=2−12(−i^+j^​+k^)−1(i^+2j^​−k^)​=(−2i^+2j^​+2k^)−(i^+2j^​−k^)=−3i^+3k^​