 Welcome to our session and let's just ask the following question. The question says, if vector a is equal to 5 by cap minus j cap minus 3k cap, vector b is equal to i cap plus 3j cap minus 5k cap, then show that vectors vector a plus vector b and vector a minus vector b are orthogonal solution. We are given that vector a is equal to 5i cap minus j cap minus 3k cap and vector b is equal to i cap plus 3j cap minus 5k cap. Now we will first find vector a plus vector b, vector a is equal to 5i cap minus j cap minus 3k cap and vector b is equal to i cap plus 3j cap minus 5k cap. Now this is equal to 5 plus 1 i cap plus minus 1 plus 3j cap plus minus 3 minus 5k cap. Vector a is equal to 6i cap plus 2j cap minus 8k cap. Now we will find vector a minus vector b, vector a is equal to 5i cap minus j cap minus 3k cap and vector b is equal to. Now here we will have minus sign 3j cap minus 5k, this is equal to 5 minus 1 is 4 minus 1 minus 3 minus 4 minus 3 plus 5 plus 2. We have learned that two non-zero vectors are orthogonal if and only if not product of these two vectors is 0. So here we have to show that vector a plus vector b and vector a minus vector b are orthogonal. That means we have to show that non product of vector a plus vector b and vector a minus vector b is 0. Now vector a plus vector b is equal to 6i cap plus 2j cap minus 8k cap and vector a minus vector b is equal to 4i cap minus 4j cap plus 2k cap. Now this is equal to 6 into 4 plus 2 into minus 4 minus 8 into 2 this is equal to 24 minus 8 minus 60 and this is equal to 0. So vector a plus vector b and vector a minus vector b is 0 therefore vector a plus vector b and a minus vector b orthogonal vectors. So this completes the session bye and take care.