If I were to find $\vec a\downarrow\vec b$, where $\vec a$ and $\vec b$ have a 90-degree angle between them, what is the result? Is it simply a zero vector, or is it undefined? If it's a zero vector, then are there any instances of $\vec a\downarrow\vec b$ that are undefined? For example at 180-degrees?
EDIT:
A follow-up question:
If the angle were 45 degrees, would $\vec a\downarrow\vec b=\vec b\downarrow\vec a$?
Best Answer
The result is perfectly well defined: the projection is the zero vector. The direction of the result is undefined, because the zero vector doesn't have a direction. This is completely analogous to the following question:
"When I subtract $x$ from $a$, is the result defined when $x = a$?"
The answer is "Sure. It's zero. But the answer doesn't have a sign, because only positive and negative numbers have a sign, but zero does not."
As for the projection of one vector on another when the angle is 45 degrees, the answer is "no". The projection of $a$ on $b$ points in the direction of $b$, but the projection of $b$ on $a$ points in the direction of $a$. You might have meant to ask "Are their lengths the same"? The answer there is "no" as well.
Consider a long vector pointing east and a short vector pointing north-east. The projection of the first on the second will be quite long, while the projection of the second on the first will be short (at least using the definition of projection that I like. You could check with your own definition: let $a = (10, 0)$ and $b = (1, 1)$.
IF the vectors have the same length and are 45 degrees apart, then the lengths of the two projections will be equal. But in fact, if the vectors have the same lengths, the lengths of their projections on each other will be equal, regardless of the angle.