{%hackmd @RintarouTW/About %}

# Geometric Language

For beginners, it's very ambiguous between coordinate(points) and vector conversion.

This is to clearify how it works.

$$
\cases{
P := point\\
V := vector\\
O := (0, 0) \in P\\
p := (p_x - 0, p_y - 0) = (p_x, p_y) \in P\\
}
$$

## Vector

$$
\begin{array}l
\vec{v} := \pmatrix{v_x\\v_y} \in V\\
\vec{u}\pm\vec{v} := \pmatrix{u_x\\u_y}\pm\pmatrix{v_x\\v_y} = \pmatrix{u_x\pm{v_x}\\u_y\pm{v_y}} \in V\\
n\vec{v} := \overbrace{\pmatrix{v_x\\v_y}+\cdots +\pmatrix{v_x\\v_y}}^n = n\pmatrix{v_x\\v_y} = \pmatrix{nv_x\\nv_y} \in V\\
\end{array}
$$

:::info
$$
\pmatrix{v_x\\v_y} = v_x\pmatrix{1\\0} + v_y\pmatrix{0\\1} = v_x\hat{x}+v_y\hat{y}
$$
:::

## Point (in Coordinate)

$$
\begin{array}l
O+\vec{p} &:= O + \pmatrix{p_x\\p_y}\\
&= (0 + p_x, 0 + p_y)\\ &= (p_x, p_y) \\
&= p \in P\\
\end{array}
$$

:::info
$$
\forall p \in P,\\
p = O + \vec{p} = \pmatrix{p_x\\p_y}
\iff p-O = \vec{p} = \pmatrix{p_x\\p_y}
$$
:::

$$
\begin{array}l
p \pm \vec{v} &= O + \vec{p} \pm \vec{v}\\
&= O + \pmatrix{p_x\\p_y} \pm \pmatrix{v_x\\v_y}\\
&= O + \pmatrix{p_x\pm v_x\\p_y\pm v_y}\\
&= (0 + (p_x\pm v_x), 0 + (p_y\pm v_y))\\
&= (p_x\pm v_x, p_y\pm v_y) \in P\\
p_2 - p_1 &= O + \vec{p_2} - (O + \vec{p_1})\\
&= O + \pmatrix{p2_x\\p2_y} - [O + \pmatrix{p1_x\\p1_y}] \\
&= \pmatrix{p2_x\\p2_y} - \pmatrix{p1_x\\p1_y}\\
&= \pmatrix{p2_x-p1_x\\p2_y-p1_y} \in V\\
\end{array}
$$

:::warning
There is no requirement for $p_1+p_2$.
Vector are not limited to 2 dimentional, it could be $n$-dimentional.
:::

###### tags: `math` `geometry`