 Hi friends I am Purva and today we will work out the following question. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are 2 into vector A plus vector B and vector A minus 3 into vector B externally in the ratio 1 is to 2. Also show that P is the midpoint of the line segment RQ. Let us begin with the solution now. Now the position vector of this point R which divides PQ externally in the ratio m is to n is given by vector OR is equal to m into vector B minus n into vector A upon m minus n and this is called section formula. We will use this formula here to find the position vector of the point R. So we have position vector of R is equal to 1 into vector A minus 3 into vector B minus 2 into 2 vector A plus vector B upon 1 minus 2. This is equal to vector A minus 3 vector B minus 2 into 2 is 4 vector A minus 2 vector B upon minus 1. This is equal to minus 3 vector A minus 5 vector B upon minus 1 which is equal to 3 vector A plus 5 vector B. So position vector of R is 3 vector A plus 5 vector B. Now we have to show that P is the midpoint of the line segment RQ. So if we show that position vector of P is equal to midpoint of RQ then we are through. Now midpoint of RQ is given by 3 vector A plus 5 vector B plus vector A minus 3 vector B upon 2. This is since we know that the midpoint of 2 vectors vector A and vector B is given by vector A plus vector B upon 2. So we have now this is equal to 4 vector A plus 2 vector B upon 2 and this is equal to now taking out 2 common numerator we get 2 into 2 vector A plus vector B upon 2. Canceling out this 2 we get. This is equal to 2 vector A plus vector B and this is equal to the position vector of P. So we have shown that midpoint of RQ is equal to position vector of P. Hence we get P is the midpoint of RQ. Hence we write our answer as the position vector of point R is 3 vector A plus 5 vector B. Hope you have understood the solution. Bye and take care.