 Hi friends, I am Purva and today we will work out the following question. If the vertices a, b, c of a triangle a, b, c are 1, 2, 3, comma minus 1, 0, 0 comma 0, 1, 2 respectively, then find angle a, b, c. Let us begin with the solution now. Now we have to find out this angle a, b, c. Now clearly this angle a, b, c is the angle between the vector b, a and vector b, c. So first we will find out vector b, a and vector b, c. Now we are given a has coordinates 1, 2, 3, b has coordinates minus 1, 0, 0 and c has coordinates 0, 1, 2. Now we have to find vector b, c. So we have vector b, c is equal to 0 i cap plus 1 j cap plus 2 k cap minus minus 1 i cap plus 0 j cap plus 0 k cap and we have this is equal to 1 i cap plus 1 minus 0 j cap plus 2 minus 0 k cap and this is equal to 1 i cap plus 1 j cap plus 2 k cap. So we have got vector b, c is equal to 1 i cap plus 1 j cap plus 2 k cap. Now we will find mod of vector b, c. So we have mod of vector b, c is equal to under root of 1 square plus 1 square plus 2 square and we have this is equal to under root of 1 plus 1 plus 4 which is equal to root 6. So we have got mod of vector b, c is equal to root 6. Now we will find vector b a. So we have vector b a is equal to 1 i cap plus 2 j cap plus 3 k cap minus minus 1 i cap plus 0 j cap plus 0 k cap and this is equal to 1 plus 1 i cap plus 2 minus 0 j cap plus 3 minus 0 k cap and this is equal to 2 i cap plus 2 j cap plus 3 k cap. So we have got vector b a is equal to 2 i cap plus 2 j cap plus 3 k cap. Now mod of vector b a is equal to under root of 2 square plus 2 square plus 3 square and we have this is equal to under root of 4 plus 4 plus 9 and this is equal to root 17. So we have got mod of vector b a is equal to root 17. Now we will find vector b c dot vector b a and we have this is equal to now vector b c is equal to i cap plus j cap plus 2 k cap dot vector b a is equal to 2 i cap plus 2 j cap plus 3 k cap and we get this is equal to now 1 into 2 gives 2 plus 1 into 2 gives 2 plus 2 into 3 gives 6 and we get this is equal to 10. So the dot product of vector b c and vector b a is equal to 10. Now since the angle theta between 2 vectors vector a and vector b is given by cos theta is equal to vector a dot vector b upon mod of vector a into mod of vector b therefore we have angle a b c between the vectors vector b c and vector b a is given by cos angle a b c is equal to vector b c dot vector b a upon mod of vector b c into mod of vector b a and we have this is equal to now vector b c dot vector b a is equal to 10. So we have 10 upon mod of vector b c is equal to root 6. So we have root 6 into mod of vector b a is equal to root 17. So we have root 17 this is equal to 10 upon root 102. So we have got cos of angle a b c is equal to 10 upon root 102 and this implies angle a b c is equal to cos inverse 10 upon root 102. So we write our answer as cos inverse 10 upon root 102. Hope you have understood the solution. Bye and take care.