Explanation of vector notation in 2D

Question

I need some explanation on vector notation in 2D. I am reading a text on optics where they express some tangential vector T as

T = [1, tan($\theta$)]

1. I do not understand that notation, component tan($\theta$) is vector slope, but for what 1 stands for ?

2. How to find the "usual" cartesian vector notation components T = ai + bj from the above notation?

Answer 1

That seems just a matrix notation with square brackets. It is just a tuple of components.

Here are some more or less usual representations: \begin{align} T &= [1, \tan(\theta)] \\ &= (1, \tan(\theta)) \\ &= \begin{pmatrix} 1 \\ \tan(\theta) \end{pmatrix}^\top \\ &= 1 i + \tan(\theta) j \\ &= 1 e_x + \tan(\theta) e_y \\ &= 1 b_1 + \tan(\theta) b_2 \\ \end{align}