 Hi friends, I am Pura and today I will help you with the following question. The two adjacent sides of a parallelogram are 2i cap minus 4j cap plus 5k cap and i cap minus 2j cap minus 3k cap. Find the unit vector parallel to its diagonal also file its area. Let us begin with the solution now. Now we are given this parallelogram abcd and we are also given these two adjacent sides. Now we have to find the unit vector parallel to its diagonal. So first we will find vector ac. Now in triangle abc we have vector ac is equal to vector ab plus vector bc and this is by triangle log this is equal to now vector ab is equal to 2i cap minus 4j cap plus 5k cap plus vector bc is i cap minus 2j cap minus 3k cap. This is equal to 2 plus 1i cap plus minus 4 minus 2j cap plus 5 minus 3k cap. So we get vector ac is equal to 3i cap minus 6j cap plus 2k cap. Now we have to find a unit vector parallel to this vector ac. So unit vector parallel to vector ac is given by n cap which is equal to vector ac upon mod of vector ac. This is equal to vector ac is 3i cap minus 6j cap plus 2k cap upon mod of vector ac is given by under root of 3 square plus minus 6 square plus 2 square this is equal to 3i cap minus 6j cap plus 2k cap upon under root of 9 plus 36 plus 4 and this is equal to 3i cap minus 6j cap plus 2k cap upon under root 49. So we have got n cap is equal to 3 upon 7i cap minus 6 upon 7j cap plus 2 upon 7k cap. Now we have to find the area of this parallelogram and area of a parallelogram is given by mod of cross product of its adjacent sites. So we get mod of vector ab cross vector bc and this is equal to mod of i cap j cap k cap and here we will write 2 minus 4 5 the coefficients of i cap j cap and k cap in vector ab and below this we will write the coefficients of i cap j cap and k cap in vector bc that is 1 minus 2 and minus 3. This is equal to i cap into minus 4 into minus 3 which is 12 minus minus 2 into 5 is minus 10 minus j cap into 2 into minus 3 which is minus 6 minus 1 into 5 which is 5 plus k cap into 2 into minus 2 which is minus 4 minus 1 into minus 4 which is minus 4. So we have this is equal to i cap into now 12 minus into minus will become plus. So we get plus 10 minus j cap into minus 6 and minus 5 gives minus 11 plus k cap into minus 4 minus into minus will become plus. So we get plus 4 this is equal to 22 i cap plus 11 j cap plus 0 k cap and we get this is equal to now taking out 11 common we get 11 into 2 i cap plus j cap. Now we have to find out mod of vector ab cross vector bc and we get this is equal to 11 into under root of 2 square plus 1 square which is equal to 11 into under root of 4 plus 1 which is equal to 11 into root 5. So we have got area of a parallelogram which is given by mod of vector ab cross vector bc is equal to 11 root 5. Hence we write our answer as the unit vector parallel to the diagonal is equal to 1 upon 7 into 3 i cap minus 6 j cap plus 2 k cap and area of this parallelogram is 11 root 5. This is our answer. Hope you have understood the solution. Bye and take care.