 Welcome to the session. Let us discuss the following question. The question says, express the vector a equals to 5i cap minus 2j cap plus 5k cap as sum of two vectors such that one is parallel to the vector v equals to 3i cap plus k cap and the other is perpendicular to vector v. With the solution, we are given that vector a is equal to 5i cap minus 2j cap plus 5k cap and vector v is equal to 3i cap plus k cap. Now we have to express vector a as sum of two vectors such that one is parallel to vector v and another is perpendicular to vector v. So vector a is equal to vector a1 plus vector a2, where the a1 is parallel to vector b, perpendicular to vector b. a1 is parallel to vector b. a1 is equal to lambda times vector b. This vector a1 is 3i cap plus 0j cap plus k cap, 3lambda a0j cap is number one. So by substituting a1 as 3lambda i cap plus 0j cap plus lambda k cap plus vector 0j cap a2 is equal to 3lambda plus 2j cap is perpendicular to vector v. Therefore this implies 5 minus 3lambda i cap minus 2j cap plus 5 minus lambda k cap. This implies 5 minus 3lambda into 3 minus 2 into 0 plus 5 minus lambda into 1 is equal to 0. This implies 15 minus 9lambda minus 0 plus 5 minus lambda is equal to 0. This implies 20 minus 10lambda is equal to 0. And this implies lambda is equal to 2. Equal to minus vector v. And we have found out that value of lambda is 2. So we have 2 into 2 vk, 0j cap plus 2 minus i cap. So this implies vector a plus 2k cap, the solution, bye.