Which item do we transpose for the dot product?

Question

I cannot find anywhere whether it's a rule to always transpose the first item in a dot product equation.
The example I have is for 2 column vectors (a and b) both have 3 rows.
Because the order matters: a$^T$$\cdot$b is NOT the same as a$\cdot$b$^T$.
Looking around at all the resources I have access to, everyone seems to transpose the first item. I just don't know why. If someone could explain it or point me in the right direction I would be very grateful!
Thank you!

Answer 1

Okay, here is what I figured out. It seems to make sense to me.
Turns out my question isn't a "valid" one as some understanding of the definition and the goal of the dot product will show.
Criteria of the dot product:

1. Same Dimension: Both vectors must have the same number of elements.
2. One-Dimensional Vectors (I didn't realise this initially): Both entities must be vectors and NOT matrices.
3. Real or Complex Numbers (not relevant to my question).
4. Order Doesn't Matter: a$\cdot$b = b$\cdot$a

The goal of the dot product (also known as the scalar product) is to provided a way to multiply two vectors together to yield a scalar value.

So, my inital question of which one should be transposed depends, as long as they are both one-dimensional vectors that have the same dimension and whichever item is being transposed will lead to a scalar value.
Thank you all for pointing me in the right direction and making me think instead of just giving me the answer.