 Hello and welcome to this session. In this session we discussed the following question that says find the position vector of a point r which divides the line joining point p and q whose position vectors are 2 vector a plus vector b and vector a minus 3 vector b respectively externally in the ratio 1 is to 2 also show that p is the midpoint of the line segment r, q. Before moving on to the solution let's discuss the section formula for external division. If suppose we have two points a and b and we are given that the position vector of the point a is vector a its position vector of the point b is vector b we have a point p which divides the line joining the points a and b that is p divides a b externally m is to n the position vector of the point p b equal to vector r is equal to vector b minus n into vector a upon n minus n is the key idea that we use in this question. Let's proceed with the solution now. We are given two points p and q and this point r divides the line joining points p and q externally in the ratio 1 is to 2. We have also given the position vectors of the points p and q and we are supposed to find the position vector of the point r. Two points p and q r is the point which divides the line joining the points p and q in the ratio 1 is to 2. We have also taken the position vectors of these points p, r and q that is we have let the position vector of point p be vector p position vector point q be vector q and the position vector of point r be vector r. We are already given the position vectors of the points p and q is equal to vector q is equal to vector a minus 3 vector b. Now, from the key idea we have the position vector of the point p the line joining the points a b externally in the ratio m is to n is given by vector r equal to vector b minus m into vector a this whole upon m minus n. Now, we are given that the point r ratio 1 is to 2 can be compared with the ratio m is to the idea we have vector r is equal to m minus 1 into the position vector of the point q which is vector a vector b which is 2 vector a plus vector b and this whole upon n minus n that is 1 minus 2. So, this is equal to vector a 4 vector a minus 2 vector b or you can say vector r is equal to minus 3 vector a minus 5 vector b is upon minus 1 or you can say this means that vector r is equal to 3 vector a plus 5 vector b. Therefore, we can say that the position vector 3 vector a plus 5 vector b show that point p line segment are line joining the points r and q in the same ratio this is what we have to show r to the ratio 1 is to 1. At the position vector of a point a as vector a position vector of a point b as vector b point p divides internally the ratio m is to have position vector of point p as vector r then is equal to this one upon m plus a using this result the position vector vector q plus 1 into which is vector r you already know vector b 1 plus 1 which is so this is equal to vector a minus 3 vector b plus 3 vector a plus 5 vector b upon 4 vector a vector b upon 2 which is equal to 2 vector a plus vector b the position vector of the midpoint of r q is 2 vector a plus vector b vector b is the vector p which is the position vector of the point p as we say that this is same as the position vector this is same as vector p and therefore we can now say this midpoint hope you understood the solution of this question.