 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 i cap plus j cap plus k cap, vector b is equal to 2 i cap minus 2 j cap plus 3 k cap, vector c is equal to 3 i cap minus 3 j cap plus 2 k cap proves that vector a dot vector b plus vector c cross vector a plus 2 vector b plus 3 vector c is equal to scalar triple product of vectors a b c. Let us proceed with the solution. We are given vector a is equal to i cap plus j cap plus k cap, vector b is equal to 2 i cap minus 2 j cap plus 3 k cap, vector c is equal to 3 i cap minus 3 j cap plus 2 k cap, then vector b plus vector c is given by 2 i cap minus j cap plus 3 k cap plus 3 i cap minus 3 j cap plus 2 k cap which is equal to 2 i cap plus 3 i cap is 5 i cap minus 2 j cap minus 3 j cap is minus 5 j cap plus 3 k cap plus 2 k cap is 5 k cap. So, vector b plus vector c is equal to 5 i cap minus 5 j cap plus 5 k cap. Now, we shall find out vector a plus 2 vector b plus 3 vector c, vector a plus 2 vector b plus 3 vector c is given by vector a is equal to i cap plus j cap plus k cap. So, we have i cap plus j cap plus k cap plus twice of vector b and vector b is given by 2 i cap minus 2 j cap plus 3 k cap. So, we have plus twice of 2 i cap minus 2 j cap plus 3 k cap plus twice of vector c and vector c is given by 3 i cap minus 3 j cap plus 2 k cap that is plus twice of 3 i cap minus 3 j cap plus 2 k cap which is equal to i cap plus j cap plus k cap plus 2 into 2 i cap that is 4 i cap plus 2 into minus 2 j cap that is minus 4 j cap plus 2 into 3 k cap that is 6 k cap plus 3 into 3 i cap that is 9 i cap 3 into minus 3 j cap that is minus 9 j cap plus 3 into 2 k cap that is plus 6 k cap which is equal to i cap plus 4 i cap plus 9 i cap that is 4 3 i cap plus j cap minus 4 j cap minus 9 j cap is minus 12 j cap plus k cap plus 6 k cap plus 6 k cap that is 13 k cap. So, we have vector a plus twice of vector b plus twice of vector c is equal to 14 i cap minus 12 j cap plus 14 k cap. So, now, we have vector b plus vector c is equal to 5 i cap minus 5 j cap plus 5 k cap and vector a plus twice of vector b plus twice of vector c is equal to 14 i cap minus 12 j cap plus 13 k cap. Now, we shall find the cross product of vector b plus vector c and vector a plus twice of vector b plus twice of vector c that is vector b plus vector c cross vector a plus twice of vector b plus twice of vector c is given by which is equal to i cap into minus 5 into 13 that is minus 65 minus of minus 12 into 5 that is minus 60 minus j cap into 5 into 14 that is 65 minus of 14 into 5 that is 70 plus k cap into minus 12 into 5 that is minus 60 minus of 14 into minus 5 that is minus 70 which is equal to i cap into minus 65 plus 60 minus j cap into minus 5 plus k cap into minus 60 plus 70 which is equal to minus 5 i cap plus 5 j cap plus 10 k cap. So, cross product of vector b plus vector c and vector a plus twice of vector b plus twice of vector c is given by minus 5 i cap plus 5 j cap plus 10 k cap. We need to prove that vector a dot vector b plus vector c cross vector a plus twice of vector b plus twice of vector c is equal to scalar triple product of vector a b c. Taking the left hand side we have vector a dot vector b plus vector c cross vector a plus twice of vector b plus twice of vector c is equal to vector a is given by i cap plus j cap plus k cap we have i cap plus j cap plus k cap dot vector b plus vector c cross vector a plus twice of vector b plus twice of vector c is given by minus 5 i cap plus 5 j cap plus 10 k cap. So, we can write minus 5 i cap plus 5 j cap plus 10 k cap which can be written as i cap dot minus 5 i cap plus j cap dot 5 j cap plus k cap dot 10 k cap which is equal to i cap into minus 5 i cap is minus 5 plus j cap into 5 j cap plus 5 plus k cap into 10 k cap is 10 which is equal to 10. So, we have the value of vector a dot vector b plus vector c cross vector a plus twice of vector b plus twice of vector c is equal to 10. Now, we shall find the scalar triple product of vector a b c which is given by the determinant of vectors a b and c. The vector a is given by i cap plus j cap plus k cap vector b is given by 2 i cap minus 2 j cap plus 3 k cap and vector c is given by 3 i cap minus 3 j cap plus 2 k cap. Scalar triple product of vector a b c is given by 1 into minus 2 into 2 that is minus 4 minus of minus 3 into 3 that is minus 9 minus 1 into 2 into 2 that is 4 minus of 3 into 3 that is 9 plus 1 into 2 into minus 3 that is minus 6 minus of 3 into minus 2 that is minus 6 which is equal to 1 into minus 4 plus 9 minus 1 into minus 5 plus 1 into minus 6 plus 6 which is equal to 1 into 5 minus 1 into minus 5 plus 1 into 0 that is given by 1 into 5 that is 5 minus 1 into minus 5 that is plus 5 plus 1 into 0 is 0 that is equal to 10. Therefore, scalar triple product of vector a b c is equal to 10. Now we know that vector a dot vector b plus vector c plus vector a plus twice of vector b plus twice of vector c is equal to 10 also scalar triple product of vector a b c is equal to 10. Therefore, we can say vector a dot vector b plus vector c plus vector a plus twice of vector b plus twice of vector c is equal to scalar triple product of vector a b c this completes our session but we enjoyed this session.