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Question:

L and M are two points with position vectors 2→a−→b and →a+2→b respectively. Write the position vector of a point N which divides the line segment LM in the ratio 2 : 1 externally.

Solution:

−−→OL=2→a−→b and −−→OM=→a+2→b
Since N divides LM in 2:1 externally, so −−→ON=1.−−→OL−1.−−→OM1−1=2(2→a−→b)−(→a+2→b)
Therefore, −−→ON=3→a−3→b