 Hi friends I am Poojwa and today we will work out the following question. Prove that the dot product of vector A plus vector B and vector A plus vector B is equal to mod of vector A square plus mod of vector B square if and only if vector A comma vector B are perpendicular given vector A is not equal to zero vector and vector B is not equal to zero vector. Let us now begin with the solution. Now here we have to prove that if vector A plus vector B dot vector A plus vector B is equal to mod of vector A square plus mod of vector B square then vector A and vector B are perpendicular that is vector A dot vector B is equal to zero and if vector A and vector B are perpendicular that is if vector A dot vector B is equal to zero then vector A plus vector B dot vector A plus vector B is equal to mod of vector A square plus mod of vector B square. Now first we will consider that vector A plus vector B dot vector A plus vector B is equal to mod of vector A square plus mod of vector B square and we will prove that vector A dot vector B is equal to zero. This implies now opening the bracket on left hand side we get vector A dot vector A plus vector A dot vector B plus vector B dot vector A plus vector B dot vector B is equal to mod of vector A square plus mod of vector B square. This implies now vector A dot vector A is equal to mod of vector A square plus vector A dot vector B plus now since vector B dot vector A is equal to vector A dot vector B so we get here vector A dot vector B plus vector B dot vector B is equal to mod of vector B square and this is equal to mod of vector A square plus mod of vector B square. Now cancelling out mod of vector A square and mod of vector B square from both the sides we get this implies vector A dot vector B plus vector A dot vector B is equal to zero and this implies 2 into vector A dot vector B is equal to zero which further implies vector A dot vector B is equal to zero. So by taking vector A plus vector B dot vector A plus vector B is equal to mod of vector A square plus mod of vector B square we have proved that vector A dot vector B is equal to zero. Now we consider vector A dot vector B is equal to zero and we have to prove that vector A plus vector B dot vector A plus vector B is equal to mod of vector A square plus mod of vector B square. So here first we will consider the left hand side and we will prove it is equal to right hand side. So we have left hand side is equal to vector A plus vector B dot vector A plus vector B. Now this is equal to vector A dot vector A plus vector A dot vector B plus vector B dot vector A plus vector B dot vector B and this is equal to now vector A dot vector A is mod of vector A square plus vector a dot vector b now since vector b dot vector a is equal to vector a dot vector b so here we will write plus vector a dot vector b plus mod of vector b square now since vector a dot vector b is equal to 0 therefore we get this is equal to mod of vector a square plus mod of vector b square and this is equal to the right hand side so by taking vector a dot vector b is equal to 0 we have proved that vector a plus vector b dot vector a plus vector b is equal to mod of vector a square plus mod of vector b square hence we write our answer as the dot product of vector a plus vector b with itself is equal to mod of vector a square plus mod of vector b square if and only if vector a comma vector b are perpendicular hope you have understood the solution bye and take care