If `bar(a),bar(b),bar(c)`are the position vectors of the points A, B, C respectively and `2bar(a)+3bar(b)-5bar(c)=bar(0)`, then find the ratio in which the point C divides line segment AB.

Explanation

Let the ration be `lambda :1`
Position vector of point (C) `vecc=(lambda veca+vecb)/(lambda+1)`
Put the value of `vecc" in "2veca +3 vecb-5vecc=0`
`2veca+3vecb-5xx((lambda veca+vecb)/(lambda+1))=0`
`rArr" "((lambda+1)(2veca+3vecb)-5lambda veca- 5vecb)/((lambda+1))=0`
`rArr 2veca lambda+3vecb lambda+2veca +3vecb-5lambda veca-5vecb=0`
`rarr" "3vecb lambda-3vec a lambda+2veca-vecb=0`
`rArr" "3lambda(vecb-veca)=2vecb-2veca`
`rArr" "3lambda =(2(vecb-veca))/((vecb-veca))`
`rArr" "(lambda)/(1)=(2)/(3)`
Hence the point C divides line segment AB in ratio `2:3`