 Hello and welcome to this session. In this session we discussed the following question which says if vector A plus vector B plus vector C is equal to 0 and magnitude of vector A is equal to 3, magnitude of vector B is equal to 5 and magnitude of vector C is equal to 7 show that the angle between vector A and vector B is 60 degrees. We know that angle between two non-zero vectors vector A and vector B is given by cos theta equal to vector A dot vector B upon magnitude of vector A into magnitude of vector B that is we have theta is equal to cos inverse vector A dot vector B upon magnitude of vector A into magnitude of vector B. This is the key idea that we use for this question. Now let's move on to the solution. Now it's given in the question that vector A plus vector B plus vector C is equal to 0. So from here we have vector A plus vector B is equal to minus vector C. Now next we have vector A plus vector B dot vector A plus vector B this would be equal to minus vector C dot minus vector C this further implies vector A dot vector A plus vector A dot vector B plus vector B dot vector A plus vector B dot vector B equal to minus vector C dot minus vector C that is vector C dot vector C. We know that vector X dot vector X is equal to magnitude of vector X the whole square and also vector A dot vector B is equal to vector V dot vector A. Since we know that scalar product or you can say the dot product of this is commutative. So using these two results this implies magnitude of vector A the whole square plus vector A dot vector B plus vector A dot vector B plus magnitude of vector B the whole square equal to magnitude of vector C the whole square. We are given that magnitude of vector A is 3. So this is 3 whole square plus 2 times vector A dot vector B plus magnitude of vector B whole square and we know the magnitude of vector B is 5. So this is 5 whole square equal to magnitude of vector C whole square that is 7 whole square since magnitude of vector C is 7. So this further implies 9 plus 2 into vector A dot vector B plus 25 equal to 49. So we get 2 times vector A dot vector B is equal to 49 minus 9 minus 25 that is 2 times vector A dot vector B is equal to 15 that is we get vector A dot vector B is equal to 15 upon 2. Now we take let theta be the angle between vector A and vector B then we have cos theta is equal to vector A dot vector B upon magnitude of vector A into magnitude of vector B that is we get cos theta is equal to vector A dot vector B that is 15 upon 2 whole upon magnitude of vector A that is 3 into magnitude of vector B that is 5 and this is equal to 15 upon 2 into 3 into 5 and 5 3 times is 15 this 3 and this 3 gets cancelled so this is equal to 1 upon 2 that is we get cos theta is equal to 1 upon 2 that is this is equal to cos 60 degrees so this implies that theta is equal to 60 degrees that is we have hence the angle between vector A and vector B is 60 degrees. So this completes the session hope you have understood the solution for this question.