Why shift the result of subtracting vectors?

Question

Why do we shift the resulting vector after subtracting any two vectors a and b?

I learned that when you add any two vectors a and b, the sum vector looks like the one that can be seen in the picture below (result could be found by adding the respective x and y coordinates of the two vectors, or any visual/geometric methods, such as the triangle or parallelogram methods).

Now since the subtraction of vector b from a can be defined as the addition of the negative (reversed) version of vector b (or a + (-b)), shouldn't the result be something like this:

Here vector v is subtracted from vector u, or u + (-v).

Instead, my book and the website that explains this concept say that the resulting vector can simply be shifted or translated so that a complete "triangle" of sorts is formed like so:

Note that the vector (u - v) has been shifted to the right now.

While I understand that the direction and magnitude of the vector haven't changed as a result of the shifting, I don't understand why we can just change it's origin or position in the space. I'm assuming all these vectors are defined by a coordinate pair, meaning that their origin is implicitly defined to be the origin (0, 0), right? The addition and subtraction of any two vectors originating from (0, 0) would therefore also start at (0, 0,) as is the case when adding two vectors. However shifting the resulting vector of subtracting two vectors results in a vector that has two non-"origin" points. What am I understanding wrongly here? I'd appreciate any clarification!

So, 1) what reasons would one have to shift such a vector and 2) why is it not mathematically incorrect to do so?

Answer 1

Answer to (2): Informally, a vector is defined by its length and its orientation, and nothing else. You can consider the vector to be located at any position that's convenient, and move it however you like — any translated version of a vector is considered to be the same vector because it retains the same length and orientation.

Answer to (1): Thinking visually helps tremendously when doing math. You might translate vectors so that the picture of a sum of two vectors or a difference of two vectors makes intuitive geometric sense. To me, the second picture of vector subtraction is easier to remember and imagine.

Answer 2

Remember we represent vectors as arrows, but they are not as such. Vectors are a way to represent a displacement! For this reason every two vectors with same magnitude and direction are indistinguishable for the majority of the mathematicians and physicists I guess. Whether you apply this vector to the origin or you shift it, the vector is the same.

Maybe it is more clear, for someone who's learning about vectors, to use your diagram, however the pro of using the other one is that in that way, you can use the OTHER diagonal of the same parallelogram to find the difference, and this is easier to remember.