 Hi and welcome to the session, let's work out the following question. The question says the vector equations of two lines are vector r is equal to i cap plus 2j cap plus 3k cap plus lambda times i cap minus 3j cap plus 2k cap and vector r is equal to 4i cap plus 5j cap plus 6k cap plus mu times 2i cap plus 3j cap plus k cap. Find the shortest distance between the above lines. So let us start with the solution to this question. We know that the shortest distance between the lines say vector r equals to vector a1 plus lambda times vector b1 and vector r is equal to vector a2 plus say mu times vector b2 is given by d is equal to mod of vector a2 minus vector a1 into vector b2 minus vector b1. Now this is vector b2 multiplied by vector b1 divided by mod of vector b1 cross vector b2. Here also we can write vector b1 cross vector b2. We call this one. Now in this question we see that this is a1, this is a2, this is b1 and this is b2. So we simply put in the values here and we get a2 minus a1 is equal to 3i cap because here we have 4i cap minus i cap. So 3i cap plus 3j cap plus 3k cap and we see that vector b1 cross vector b2 is equal to determinant i cap j cap k cap 1 minus 3 2 2 3 1. This is equal to i cap into minus 3 minus 6 minus j cap into 1 minus 4 plus k cap into 3 plus 6. This is equal to minus 9i cap plus 3j cap plus 9k cap. Therefore now we have also we see that vector a2 minus vector a1 multiplied by vector b1 cross vector b2 is equal to, now this is a2 minus a1 or we can write here vector a2 minus vector a1. So we have 3i cap plus 3j cap plus 3k cap multiplied by minus 9i cap plus 3j cap plus 9k cap and this is equal to 3 into minus 9 is minus 27 plus 3 into 3 is 9 plus 3 into 9 is 27. 27 gets cancelled with minus 27 and we have 9 and we see that mod of vector b1 cross vector b2 is equal to square root of a1 that is square of minus 9 plus square of 3 that is 9 plus square of 9 that is 81 that sums up to square root of 171. This is equal to 3 into square root of 19. So we have now d will be equal to mod of 9 divided by 3 root 19 and this is equal to 3 divided by root 19. So our answer to this question is 3 divided by square root of 19. I hope that you understood the solution and enjoyed the session. Have a good day.