 Hello and welcome to the session. In this session we discussed the following question which says prove that the points A, B, C with position vectors minus 7 vector A plus 2 vector B minus 3 vector C minus 5 vector A plus 3 vector B minus 8 vector C and minus vector A plus 5 vector B minus 18 vector C respectively are collinear whatever be vector A vector B vector C. Let's move on to the solution now. We are given that a point A has position vector there so we say position vector of point A is equal to minus 7 vector A plus 2 vector B minus 3 vector C then position vector of point B is equal to minus 5 vector A plus 3 vector B minus 8 vector C and the position vector of point C is equal to minus vector A plus 5 vector B minus 18 vector C then vector AB is given by position vector of point B minus the position vector of point A so this is equal to minus 5 vector A plus 3 vector B minus 8 vector C minus minus 7 vector A plus 2 vector B minus 3 vector C and this would be equal to 2 vector A plus vector B minus 5 vector C this is vector AB. Now in the same way we can find out vector AC which is equal to the position vector of point C minus position vector of point A so this is equal to minus vector A plus 5 vector B minus 18 vector C minus minus 7 vector A plus 2 vector B minus 3 vector C so this is equal to 6 vector A plus 3 vector B minus 15 vector C this is vector AC. Next we find out the cos product of the vectors AB and AC so vector AB cos vector AC is equal to 2 vector A plus vector B minus 5 vector C cross 6 vector A plus 3 vector B minus 15 vector C or we can write this as 2 vector A plus vector B minus 5 vector C cross 3 into 2 vector A plus vector B minus 5 vector C or you can say that we get vector AB cross vector AC is equal to vector D cross 3 into vector D where we have taken vector D to be equal to 2 vector A plus vector B minus 5 vector C. Now since we know that vector D cross vector D is equal to a 0 vector therefore we say that vector AB cross vector AC is equal to a 0 vector. Hence we say that vector AB and vector AC are parallel vectors having a common point A and thus points AB and CR collinear. This is what we were supposed to prove so this can be easy session. Hope you have understood the solution of this question.