 Hello and welcome to the session. In this session we are going to discuss the following question which says that evaluate vector A cross vector B dot vector C plus vector A cross vector C dot vector B where vector A is equal to i cap plus 2 j cap plus k cap vector B is equal to 2 i cap plus 3 j cap plus k cap vector C is equal to 3 i cap plus 2 j cap plus 3 k cap Let us proceed with the solution. We are given vector A as i cap plus 2 j cap plus k cap vector B as 2 i cap plus 3 j cap plus k cap and vector C as 3 i cap plus 2 j cap plus 3 k cap Now we shall find out vector A cross vector B which is given by the determinant of vector A and vector B that is i cap into 2 into 1 that is 2 minus of 3 into 1 that is 3 minus j cap into 1 into 1 that is 1 minus of 2 into 1 that is 2 plus k cap into 1 into 3 that is 3 minus of 2 into 2 that is 4 which is equal to i cap into 2 minus 3 that is minus 1 which is minus i cap minus j cap into 1 minus 2 that is minus 1 which is equal to plus j cap plus k cap into minus 1 that is minus k cap So vector A cross vector B is equal to minus i cap plus j cap minus k cap Now we find vector A cross vector B dot vector C which is equal to vector A cross vector B is given by minus i cap plus j cap minus j cap dot vector C is given by 3 i cap plus 2 j cap plus 3 k cap that is 3 i cap plus 2 j cap plus 3 k cap So we have minus i cap into 3 i cap that is minus 3 plus j cap into 2 j cap that is plus 2 minus k cap into 3 k cap that is minus 3 which is equal to minus 4 vector A cross vector B dot vector C is equal to minus 4 Now we shall find vector A cross vector C which is given by vector A is equal to i cap plus 2 j cap plus k cap and vector C is equal to 3 i cap plus 2 j cap plus 3 k cap vector A cross vector C is given by the determinant of vector A and vector C which is equal to i cap into 2 into 3 that is 6 minus of 2 into 1 that is 2 minus j cap into 1 into 3 that is 3 minus of 3 into 1 that is 3 plus k cap into 1 into 2 that is 2 minus of 3 into 2 that is 6 which is equal to i cap into 6 minus 2 that is 4 minus j cap into 3 minus 3 that is 0 plus k cap into 2 minus 6 that is minus 4 that is 4 i cap minus 0 j cap minus 4 k cap So vector A cross vector C is given by 4 i cap minus 0 j cap minus 4 k cap Now vector A cross vector C dot vector B is given by vector A cross vector C is 4 i cap minus 0 j cap minus 4 k cap dot vector B vector B is equal to 2 i cap plus 3 j cap plus k cap that is 2 i cap plus 3 j cap plus k cap which is equal to 4 i cap into 2 i cap that is 8 minus 0 j cap into 3 j cap that is minus 0 minus 4 k cap into k cap is minus 4 So we have 8 minus 4 that is 4 So vector A cross vector C dot vector B is equal to 4 We need to find the value of the expression vector A cross vector B dot vector C plus vector A cross vector C dot vector B that is vector A cross vector B dot vector C plus vector A cross vector C dot vector B Now we have vector A cross vector B dot vector C is equal to minus 4 and vector A cross vector C dot vector B is equal to 4 So we can write vector A cross vector B dot vector C plus vector A cross vector C dot vector B as minus 4 plus 4 which is equal to 0 Therefore the value of the expression vector A cross vector B dot vector C plus vector A cross vector C dot vector B is equal to 0 which is the required answer This completes our session hope you enjoyed this session