 Good morning friends, I am Poojwa and today we will work out the following question. Show that each of the given three vectors is a unit vector and we are given 1 upon 7 into 2 i cap plus 3 j cap plus 6 k cap, 1 upon 7 into 3 i cap minus 6 j cap plus 2 k cap and 1 upon 7 into 6 i cap plus 2 j cap minus 3 k cap also show that they are mutually perpendicular to each other. Let us now begin with the solution. Now let us denote vector a is equal to 1 upon 7 into 2 i cap plus 3 j cap plus 6 k cap, vector b is equal to 1 upon 7 into 3 i cap minus 6 j cap plus 2 k cap and vector c is equal to 1 upon 7 into 6 i cap plus 2 j cap minus 3 k cap. Now we have to show that each of the vectors a, b and c is the unit vector. So for that we will show that magnitude of all the three vectors is equal to 1. So first we consider mod of vector a and this is equal to under root of 2 square plus 3 square plus 6 square upon 7 square and we get this is equal to under root of 2 square is equal to 4 plus 3 square is equal to 9 plus 6 square is equal to 36 upon 7 square which is equal to 49 and we get this is equal to under root 49 upon 49 and we get this is equal to 1. So we have got mod of vector a is equal to 1. Now we find out mod of vector b and this is equal to under root of 3 square plus minus 6 square plus 2 square upon 7 square and we get this is equal to under root of 9 plus 36 plus 4 upon 49 and we get this is equal to under root of now 9 plus 36 plus 4 gives 49. So we have 49 upon 49 and we get this is equal to 1. So we have got mod of vector b is equal to 1. Now finally we find mod of vector c and we have mod of vector c is equal to under root of 6 square plus 2 square plus minus 3 square upon 7 square and we get this is equal to under root of 36 plus 4 plus 9 upon 49 and we get this is equal to under root of 49 upon 49 which is equal to 1. So we have got mod of vector c is equal to 1. So this shows that vector a, vector b and vector c are unit vectors because the magnitude of each of them is equal to 1. Now we have to show that vector a, vector b and vector c are mutually perpendicular to each other. So for that first we will find vector a dot vector b, vector b dot vector c and vector c dot vector a. So now vector a dot vector b is equal to 1 upon 7 into 2 i cap plus 3 j cap plus 6 k cap dot 1 upon 7 into 3 i cap minus 6 j cap plus 2 k cap and we get this is equal to 1 upon 7 into 1 upon 7 gives 1 upon 49 into 2 into 3 plus 3 into minus 6 plus 6 into 2 and we get this is equal to 1 upon 49 into 2 into 3 gives 6 3 into minus 6 gives minus 18 plus 6 into 2 gives 12 and this implies vector a dot vector b is equal to 0 which implies vector a is perpendicular to vector b. Now we will show vector b is perpendicular to vector c. So for that first we will find vector b dot vector c. So vector b dot vector c is equal to 1 upon 7 into 3 i cap minus 6 j cap plus 2 k cap dot 1 upon 7 into 6 i cap plus 2 j cap minus 3 k cap and we get this is equal to 1 upon 49 into 3 into 6 which is 18 minus 6 into 2 which is equal to minus 12 and 2 into minus 3 which is equal to minus 6 and we get this implies vector b dot vector c is equal to 0 because minus 12 minus 6 gives minus 18 and 18 minus 18 is equal to 0. So we have got vector b dot vector c is equal to 0 which further implies vector b is perpendicular to vector c. Now finally we will show vector c is perpendicular to vector a. So first we will find vector c dot vector a and this is equal to 1 upon 7 into 6 i cap plus 2 j cap minus 3 k cap dot 1 upon 7 into 2 i cap plus 3 j cap plus 6 k cap and we get this is equal to 1 upon 49 into 6 into 2 plus 2 into 3 plus minus 3 into 6 and we get this is equal to 1 upon 49 into 6 into 2 is 12 plus 2 into 3 is 6 plus minus 3 into 6 gives minus 18 and we get this implies vector c dot vector a is equal to 0 which further implies vector c is perpendicular to vector a. Hence we have shown that vector a vector b and vector c are unit vectors and we have also shown that these three vectors are mutually perpendicular to each other. This is our answer hope you have understood the solution bye and take care