My professor gave the following theorem today:
Let A and B be vectors. Then A * B =0 if and only if A and B are perpendicular.
Now, in the proof for this my professor gave the following reasoning in the case that A or B are the zero vector, in the "iff direction" of if 0 => perpendicular:
|A| = 0 or |B| = 0 => A=0 or B=0 => Give A are direction perp. To B or give B a direction perp to A => A is perp. To B.
I understand that we can attribute any direction to the zero vector, but does anyone else feel it is a bit sketchy to "choose" A to be perp. to B in order to prove that A is perp to B?
In other words.. the z ero vector's direction is anything. I don't have to "choose" it to be perpendicular to anything for the dot product to be zero. Therefore, as the theorem reads,"if and only if A and B are perpendicular" is a bit... not "wrong" but at the same time, could be more accurate.