mathematics extension 2

The question basically is: Find the shortest distance between lines r = i+2j+3k + lambda (2i+j+4k) and r = 2i+4j+5k + mu (3i+4j+5k). I don't get these sort of questions and I really need help. Can someone give me a solution that doesn't involve the use of matrices please.

I got the RHS of the inequality just dont understand why 1 is less than nth root of n. Should it not be less than or equal to?

Is this how I should complete my proof: By way of contradiction assume that 2k is the largest even integer. Now consider (2k)! (2k)! = (2k)(2k-1)(2k-2).......(k)(k-1)(k-2)......(2)(1) = 2[k(2k-1)(2k-2).......(k)(k-1)(k-2)......(2)(1)] = 2p, which is also an even integer. This contradicts...

The question was asking: Prove the following statement using either direct or contrapositive proof: If n is an integer then 4 does not divide n^2-3 Here is my working out: let n^2 - 3 = 4m By way of contradiction assume n^2 - 3 is rational, ie; n^2 - 3 = a/b (BTW in the funky looking...

For the following question when I am trying to prove the n = k+1 case am I allowed to substitute it in the first line of the question and also in the second line of the question? What I am trying to say is am i allowed to assume: