Where's the freakin' rabbit?

I was teaching my linear algebra students on Wednesday about vector spaces, spanning sets and linear independence. I spent the entire class period working a single problem and making connections between many of the things we had been studying. To demonstrate how useful and correct my methods were, I asked my students to suggest a vector from three-space, and then I proceeded to do the calculations to show just how everything magically works out. Only, with two minutes left in the class, it became clear to everyone, including me, that the numbers were coming out all wrong. It was like building up to the moment when you pull the rabbit out of the magic hat, only to discover that the rabbit wasn't there anymore. Or like doing a magic card trick that goes totally bust:

Me: Pick a card, any card.

Audience member: (Selects a card).

Me: (In a theatrical voice) So your card was....(drum roll)...the ace of spades.

Audience member: Uh...(embarrassing pause)...no.

Me: Oh, crap.

It only took me about 10 minutes in my office after class to figure out what went wrong. I simply forgot to change one sign. 'Hah,' I thought to myself, 'I bet even Newton missed a sign or two in his day.' By the next class period, I was ready with all my computations double-checked. Sure enough, I could make all the methods work for the vector they had suggested. And then just to prove that I had complete confidence that this method would work for any other vector they might give me, I said in my most authoritative voice, "This method would most certainly work for any other vector you'd care to choose." And then to drive the point home even further, I moved on to the next topic in the lesson.