Recently, I have seen this notation: $\vec{a}= l·\begin{pmatrix} \cos(270) \\ \sin(270) \\ \end{pmatrix}= l·\begin{pmatrix} 0 \\ -1 \\ \end{pmatrix}$
Where the coordinates of the vector are expressed with trigonometric functions, and then scaled to the proper length afterwards. I don't get how the trigonometric function can be used as coordinates for the vector. I hope you can explain this for me.
I drew a sketch of the situation: Vector sketch
Thanks in advance!
On
Use trigonometric function can be useful to parametrize a line through the origin and with slope $\theta$ with respect to $x$ axis that is
$$\vec{a}= l·\begin{pmatrix} \cos \theta \\ \sin \theta \\ \end{pmatrix}$$
with $l\in \mathbb{R}$ parameter.
Note that in your example it should be
$$\vec{a}= l·\begin{pmatrix} \cos(270) \\ \sin(270) \\ \end{pmatrix}= l·\begin{pmatrix} 0 \\ \color{red}{-1} \\ \end{pmatrix}$$
Vectors have real numbers as their coordinates. Here $\cos(270)$ and $\sin(270)$ are names of two particular real numbers. We can get other names for this numbers if we evaluate $\cos(270)$ and $\sin(270)$. This names would be $0$ and $-1$ respectively.
Sometimes functions are used to name things.
You can think of $\cos(270)$ as "that real number which is cosine of 270". The same applies to $\sin(270)$.
So your vector becomes
$$\vec{a}= l·\begin{pmatrix} \text{"that real number which is cosine of 270"} \\ \text{"that real number which is sine of 270"} \\ \end{pmatrix}$$
Multiplication by $l$ multiplyes both coordinates and you get an effect of scaling.
Added:
Why are there sine and cosine?
The pair $(cos(x), sin(x))$ is there to select some point on a unit circle. A vector with such coordinates will have length $1$ and, as you can imagine, can point to some direction. Then this vector is scaled by $l$ to get desired length without changing its orientation.