 Good morning friends, I am Purva and today we will discuss the following question. If vector A, vector B and vector C are unit vectors such that vector A plus vector B plus vector C is equal to zero vector then find the value of vector A dot vector B plus vector B dot vector C plus vector C dot vector A. Let us now begin with the solution. Now we are given that vector A plus vector B plus vector C is equal to zero vector. Now we can write this as this implies vector A plus vector B is equal to minus vector C. Now taking the dot product with vector A on both the sides we get vector A dot vector A plus vector B is equal to vector A dot minus vector C and we have this implies. Now opening the bracket we get vector A dot vector A which is equal to mod of vector A square plus vector A dot vector B is equal to vector A dot minus vector C and this implies mod of vector A square plus vector A dot vector B plus vector A dot vector C is equal to zero and this implies. Now since A is a unit vector therefore we have mod of vector A is equal to one so here we get one square which is equal to one so we have one plus vector A dot vector B plus vector A vector C is equal to zero and we mark this as equation one. Now again we consider vector A plus vector B plus vector C is equal to zero vector and we can write this as this implies vector A plus vector B is equal to minus vector C. Now taking dot product with vector B on both the sides we get vector B dot vector A plus vector B is equal to vector B dot minus vector C and we get this implies vector B dot vector A plus vector B dot vector B which is equal to mod of vector B square is equal to minus of vector B dot vector C and this implies vector B dot vector A plus mod of vector B square plus vector B dot vector C is equal to zero. This implies now we can write vector B dot vector A as vector A dot vector B and this is since vector B dot vector A is equal to vector A dot vector B plus now since B is a unit vector therefore we have mod of vector B is equal to one so here we get plus one plus vector B dot vector C is equal to zero and we mark this as equation two. Again we are given that vector A plus vector B plus vector C is equal to zero vector now we can write this as this implies vector A plus vector B is equal to minus vector C taking the dot product with vector C on both the sides we get vector C dot vector A plus vector B is equal to vector C dot minus vector C and this implies now opening the bracket we get vector C dot vector A plus vector C dot vector B is equal to minus of vector C dot vector C. This further implies vector C dot vector A plus vector C dot vector B plus vector C dot vector C is equal to zero and this implies vector C dot vector A plus vector C dot vector B plus now vector C dot vector C can be written as mod of vector C square is equal to zero and this implies vector C dot vector A plus vector C dot vector B plus now since C is given to be a unit vector therefore we have mod of vector C is equal to one so here we get plus one which is equal to zero and we mark this as equation three. Now adding equation one two and three we get two into vector A dot vector B plus vector B dot vector C plus vector C dot vector A plus three is equal to zero. Now this implies two into vector A dot vector B plus vector B dot vector C plus vector C dot vector A is equal to minus three and this further implies vector A dot vector B plus vector B dot vector C plus vector C dot vector A is equal to minus three upon two so we have got the value of vector A dot vector B plus vector B dot vector C plus vector C dot vector A as minus three upon two hence we write our answer as minus three upon two hope you have understood the solution. Bye and take care.