# Could a vector with complex components have a negative magnitude?

I don't have too much of a math background; I'm in calculus AB and AP physics. In my physics class, the teacher said that, to cool something below absolute zero, you would have to slow the particles until they're not moving, then slow them even more. That got me thinking, if you could theoretically give particles velocity with components that contain imaginary numbers, could you create a vector with negative magnitude? I've tried to find some combination that would make the math work, but I couldn't find a solution. However, I only did try in three dimensions, and I don't really have a good methodology to approaching this problem. I've asked my physics teacher and calculus teacher, and neither knew.

• Thanks for the answers everyone! Interesting stuff, and words of encouragement nice to hear. I'll accept an answer soon – Phoenix Mar 12 '18 at 3:29