 Hello and welcome to the session. In this session we are going to discuss the following question which says that if vector A is equal to 2i cap minus 2j cap minus k cap, vector B is equal to i cap plus 2j cap plus 2k cap, vector C is equal to 3i cap minus 4j cap plus 2k cap, then proves that scalar triple product of vectors A plus B, B plus C, C plus A is equal to twice of scalar triple product of vectors A, B, C. Let us proceed with the solution. We are given vector A as 2i cap minus 2j cap minus k cap, vector B as i cap plus 2j cap plus 2k cap, vector C as 3i cap minus 4j cap plus 2k cap. Now vector A plus vector B is given by 2i cap minus 2j cap minus k cap plus i cap plus 2j cap plus 2k cap which is equal to 2i cap plus i cap that is 3i cap minus 2j cap plus 2j cap that is 0j cap minus k cap plus 2k cap that is plus k cap, vector B plus vector C is given by i cap plus 2j cap plus 2k cap plus 3i cap minus 4j cap plus 2k cap which is equal to i cap plus 3i cap that is 4i cap plus 2j cap minus 4j cap that is minus 2j cap plus 2k cap plus 2k cap that is plus 4k cap. Now vector C plus vector A is given by 3i cap minus 4j cap plus 2k cap plus 2i cap minus 2j cap minus k cap which is equal to 3i cap plus 2i cap that is 5i cap minus 4j cap minus 2j cap that is minus 6j cap plus 2k cap minus k cap is plus k cap. Now we have vector A plus vector B is given by 3i cap plus 0j cap plus k cap vector B plus vector C is given by 4i cap minus 2j cap plus 4k cap vector C plus vector A is given by 5i cap minus 6j cap plus k cap. Now we shall find scalar triple product of vector A plus B B plus C C plus A which is given by the determinant of vector A plus B B plus C C plus A equal to 3 into minus 2 into 1 that is minus 2 minus of minus 6 into 4 that is minus 24 minus 0 into 4 into 1 that is 4 minus of 5 into 4 that is 20 plus 1 into 4 into minus 6 that is minus 24 minus of 5 into minus 2 that is minus 10 which is equal to 3 into minus 2 plus 24 minus 0 into minus 16 plus 1 into minus 24 plus 10 that is 3 into minus 2 plus 24 that is 22 minus 0 into minus 16 is 0 plus 1 into minus 24 plus 10 that minus 14 that is 66 minus 14 which is equal to 52 so scalar triple product of vectors a plus b, b plus c, c plus a is equal to 52 now we shall find scalar triple product of vectors a, b, c where vector a is given by 2i cap minus 2j cap minus j cap vector b is given by i cap plus 2j cap plus 2k cap vector c is given by 3i cap minus 4j cap plus 2k cap so scalar triple product of vectors a, b, c is given by the determinant of vectors a, b and c which is equal to 2 into 2 into 2 that is 4 minus of minus 4 into 2 that is minus 8 minus of minus 2 into 1 into 2 that is 2 minus of 3 into 2 that is 6 minus 1 into 1 into minus 4 that is minus 4 minus of 3 into 2 that is 6 which is equal to 2 into 4 plus 8 plus 2 into 2 minus 6 is minus 4 minus 1 into minus 4 minus 6 is minus 10 which is equal to 2 into 12 plus 2 into minus 4 is minus 8 minus 1 into minus 10 is plus 10 that is 24 minus 8 plus 10 which is equal to 26. So, we have scalar triple product of vectors A, B, C is equal to 26. So, twice of scalar triple product of vectors A, B, C can be written as 2 into 26 that is 52. So, we have scalar triple product of vectors A plus B, B plus C, C plus A is equal to 52 and twice of scalar triple product of vectors A, B, C is equal to 52. Therefore, scalar triple product of vectors A plus B, B plus C, C plus A is equal to twice of scalar triple product of vectors A, B, C. This completes our session. Hope you enjoyed this session.