 Hi friends, I am Puruwa and today we will discuss the following question show that mod of vector A into vector B plus mod of vector B into vector A is perpendicular to mod of vector A into vector B minus mod of vector B into vector A for any two non-zero vectors vector A and vector B. Suppose we have two vectors vector A and vector B then we say that these two vectors are perpendicular to each other if their dot product is equal to zero that is we have vector A dot vector B is equal to zero. So this is the key idea behind our question. Let us now begin with the solution. Now we have to show that these two vectors are perpendicular to each other that is we have to show that dot product of these two vectors is equal to zero. So we consider mod of vector A into vector B plus mod of vector B into vector A dot mod of vector A into vector B minus mod of vector B into vector A and we will show that this is equal to zero and we have this is equal to now we can write this as mod of vector A into vector B dot mod of vector A into vector B minus mod of vector A into vector B dot mod of vector B into vector A plus mod of vector B into vector A dot mod of vector A into vector B minus mod of vector B into vector A dot mod of vector B into vector A. Now using distributive law we can write this as this is equal to mod of vector A square into vector B dot vector B minus mod of vector A into mod of vector B into vector B dot vector A plus mod of vector B into mod of vector A into vector A dot vector B minus mod of vector B square into vector A dot vector A and this is by distributive law now this is equal to mod of vector A square vector B dot vector B can be written as mod of vector B square minus mod of vector A into mod of vector B now since we know that vector B dot vector A is equal to vector A dot vector B so here we can write vector A dot vector B plus mod of vector A into mod of vector B into vector A dot vector B minus mod of vector B square into now vector A dot vector A can be written as mod of vector A square and this is since we know that vector R dot vector R is equal to mod of vector R square now cancelling out these two terms we get this is equal to mod of vector A square into mod of vector B square minus now we can write this as mod of vector A square into mod of vector B square now these two terms will also cancel out and we will get this is equal to zero so we have got the dot product of these two vectors is equal to zero hence we write our answer as mod of vector A into vector B plus mod of vector B into vector A is perpendicular to mod of vector A into vector B minus mod of vector B into vector A this is our answer hope you have understood the solution bye and take care