Differences between $P_{|BO}=(x_P, y_P,z_P)$ and $P_{|BO}=(x_P \quad y_P \quad z_P )^T$?

Question

I know that they represent points in the space, but what are the meaning of the different notations?

Answer 1

The former represents the element as a $3$-tuple, and the latter as a column vector. It's pretty common to represent a column vector with an $n$-tuple in linear algebra, for convenience, since in the finite dimensional setting, matrices act on column vectors, and linear maps act on $n$-tuples.