Find the position vector of a point R which divides the line segment joining P and Q whose position vectors are `2 vec a+ vec b\ a n d\ vec a-4 vec b ,` externally in the ratio 1:2, also show that P is the midpoint of the line segment RQ.

Explanation

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`vec{R}=frac{2 times vec{P}-1 times vec{Q}}{2-1} `

`=4 vec{a}+6 vec{b}-vec{a}+3 vec{b}=3 vec{a}+9 vec{b}`

Mid-point of Q ,`vec{R}=frac{vec{Q}+vec{R}}{2}=frac{(vec{a}-3 vec{b}+(3 vec{a}+9 vec{b}))}{2} `

`=2 vec{a}+3 vec{b}=vec{P}`