I have no idea where to even begin, this “having to logically prove things” seems unnatural and evil to me, but is apparently expected of you in this class.
These are the two questions.
Let u and v be vectors in 3-space. Prove that ||u+v||²+ ||u-v||² = 2||u||²+ 2||v||².
And
Prove or disprove: Let u, v and w be vectors in 3-space. Then uX(vXw)= (uXv)Xw
BTW, I am past the point where you have to hand in homework to be graded. So please, keep the “we won’t do your homework for you” to a minimum. I really am unfamiliar (unpracticed, more importantly) in mathematical proofs and am not sure where to even begin. It is always so painfully obvious when the professor writes it out.
For the first one, remember that ||u||[sup]2[/sup] = u * u, where * denotes the dot product, and that (a + b) * (c + d) = a * c + a * d + b * c + b * d.
For the second, just use the definition of X. Sure, it’s a lot of algebra, but it’s not as bad as it seems.
BTW, how do you make all those nice exponents and bold letters? I will use what I know, which is tex code. For the first,
||u +v||^2 = (u+v).(u+v) = u.u + v.v + 2u.v and
||u - v||^2 = (u-v).(u - v) = u.u + v.v - 2u.v
and when you add them you get 2u.u + 2v.v, which is what you wanted. For the second, try i x (i x k) = i x (-j) = -k, while (i x i) x k = 0.
The first is called the parallelogram law. In simple geometric terms it says that the sum of the squares of the diagonals of a parallelogram is the sum of the squares of the sides.
I’ll give my impressions on how to go about proving stuff.
Sometimes there’s a trick to it, and you do or don’t spot it. You just have to live with that. Unfortunately some important results have a ‘trick’ and you might well have to learn them for an exam.
More usefully, multiply stuff out, or factorise stuff until you recognise something. Look for disguised products (as ||a||^2 above), etc.
Try working backwards from the result. Make sure that you rewrite it all in the correct order afterwards. (Proof by contradiction is similar to this.)
This isn’t helping. Can someone for whom it’s not 2am continue?
When the question says “prove or disprove”, you first have to figure out whether it’s true or false, so you know what you’re aiming for. If it’s false, then you can usually disprove it by finding a counterexample. This is often easy enough that it’s worthwhile to start by just looking for counterexamples. If one of them doesn’t work, you’re done. If you try a variety of things and they all work, then it’s probably true, so you should try to prove it.
Make your (potential) counterexample something simple, such as basis unit vectors.