 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 3i cap plus 2j cap plus 4k cap vector b is equal to minus 2i cap minus 2j cap minus 2k cap vector c is equal to 4i cap minus 2j cap plus 5k cap vector d is equal to i cap minus j cap plus k cap a find scalar product of 4 vectors b find vector product of 4 vectors. Let us proceed with the solution. We are given vector a as 3i cap plus 2j cap plus 4k cap vector d as minus 2i cap minus 2j cap minus 2k cap vector c as 4i cap minus 2j cap plus 5k cap and vector d as i cap minus j cap plus k cap. Now we shall find the scalar product of the 4 vectors which is given by vector a cross vector b dot vector c cross vector d. Now we shall find the cross product of vector a and vector b vector a is given by 3i cap plus 2j cap plus 4k cap and vector d is given by minus 2i cap minus 2j cap minus 2k cap cross product of vector a and b is given by the determinant of vector a and b which is equal to i cap into 2 into minus 2 that is minus 4 minus off minus 2 into 4 that is minus 8 minus j cap into 3 into minus 2 that is minus 6 minus off minus 2 into 4 that is minus 8 plus k cap into 3 into minus 2 that is minus 6 minus off minus 2 into 2 that is minus 4 which is equal to i cap into minus 4 plus 8 minus j cap into minus 6 plus 8 plus k cap into minus 6 plus 4 which is equal to 4i cap minus 2j cap minus 2k cap. Therefore the cross product of vector a and vector b is equal to 4i cap minus 2j cap minus 2k cap. Now we shall find out the cross product of vector c and vector d. Vector c is given by 4i cap minus 2j cap plus 5k cap and vector d is given by i cap minus j cap plus k cap. Cross product of vector c and vector d is given by the determinant of vector c and d which is equal to i cap into minus 2 into 1 that is minus 2 minus off minus 1 into 5 that is minus 5 minus j cap into 4 into 1 that is 4 minus off 1 into 5 that is 5 plus k cap into 4 into minus 1 that is minus 4 minus off 1 into minus 2 that is minus 2 which is equal to i cap into minus 2 plus 5 minus j cap into minus 1 plus k cap into minus 4 plus 2 which is equal to 3i cap plus j cap minus 2k cap. Therefore cross product of vector c and vector d is equal to 3i cap plus j cap minus 2k cap. We know that the scalar product of the 4 vectors is given by vector a cross vector d dot vector c cross vector d. Vector a cross vector d is given by 4i cap minus 2j cap minus 2k cap and vector c cross vector d is given by 3i cap plus j cap minus 2k cap. So vector a cross vector d dot vector c cross vector d is equal to 4i cap minus 2j cap minus 2k cap dot 3i cap plus j cap minus 2k cap which is equal to 4i cap into 3i cap is 12 minus 2j cap into j cap is minus 2 minus 2k cap into minus 2k cap is plus 4 which is equal to 14. Therefore scalar product of the 4 vectors dc and d is given by 14 which is the required answer. Now we shall find the vector product of the 4 vectors that is vector a cross vector b cross vector c cross vector b and we know that vector a cross vector b is given by 4i cap minus 2j cap minus 2k cap and vector c cross vector d is given by 3i cap plus j cap minus 2k cap. So vector A cross vector B cross vector C cross vector D is given by the determinant of vector A cross vector B and vector C cross vector D which is equal to i cap into minus 2 into minus 2 that is 4 minus of 1 into minus 2 that is minus 2 minus j cap into 4 into minus 2 that is minus 8 minus of 3 into minus 2 that is minus 6 plus k cap into 4 into 1 that is 4 minus of 3 into minus 2 that is minus 6 which is equal to i cap into 4 plus 2 minus j cap into minus 8 plus 6 plus k cap into 4 plus 6 which is equal to 6 i cap plus 2 j cap plus 10 k cap. Therefore vector product 54 vectors B, C and D is given by 6 i cap plus 2 j cap plus 10 k cap which is the required answer. This completes our session. Hope you enjoyed this session.