---
tags: Math, Vector Algebra
title: Vector Alebra
author:
license:
---

$\Huge \text{Vector Algebra}$

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:::info
> 🔗 Link:
> 📌 Tags:
> 📋 Content:
> [TOC]
>
> [color=blue]
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## <center> A. Giới thiệu </center>

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### <center> A.1) Đối tượng độc giả </center>

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### <center> A.2) Thông tin khác </center>

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## <center> B. Không gian $\mathbb{R}^2$ </center>

Kí hiệu $\mathbb{R}^2$ là tập hợp:

$$
\mathbb{R}^2=\{\mathbf u=(x;y)\mid x,y\in\mathbb{R}\}
$$

Khi đó $\mathbb{R}^2$ là không gian vector

Mọi $u =(x; y) \in \mathbb{R}^2$ mô phỏng điểm $A(x; y)$ trên mặt phẳng $Oxy$

Để cho tiện, ta sẽ định nghĩa các biến sau
> Góc degree giữa $2$ vector $\phi \in \mathbb{R}$
> Góc radian giữa $2$ vector $\theta \in \mathbb{R}$
> Đại số vô hướng $\alpha, \beta \in \mathbb{R}$
> Sai số $\varepsilon \in \mathbb{R}$
> Scalar $u_x, u_y, v_x, v_y, w_x, w_y \in \mathbb{R}$
> Vector $\mathbf u = (u_x; u_y), \mathbf v = (v_x; v_y), \mathbf w = (w_x; w_y) \in \mathbb{R}^2$
> [color=green]

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### <center> B.1) Tính chất vector cơ bản </center>

:::info
**Definition**:
$$
\text{vector }\mathbf u=(u_x;\ u_y)\in\mathbb{R}^2\\
\text{scalar }u_x, u_y \in \mathbb{R}
$$
:::

> **Zero Vector**:
$$
\mathbf0=(0;0)\\
\exists!\mathbf u=\mathbf 0\to\|\mathbf u\|=0
$$
> **Unit Vector**: $\forall\mathbf u\neq\mathbf 0$
$$
\widehat{\mathbf u} = (\sin(\theta);\ \cos(\theta)) = \frac{\mathbf u}{\|\mathbf u\|}
$$
> **Basis Vector**
$$
\mathbf i=\widehat{\mathbf i}=(1;0)\\
\mathbf j=\widehat{\mathbf j}=(0;1)\\
$$
> **Additive Identity**:
$$
0 + \mathbf{u} = \mathbf{u} + 0 = \mathbf{u}
$$
> **Additive Inverse**:
$$
\mathbf{u} + (-\mathbf{u}) = \mathbf{0}
$$
> **Double Negation**:
$$
-(-\mathbf{u}) = \mathbf u
$$
> **Associativity**:
$$
(\alpha\beta)\mathbf u=\alpha(\beta\mathbf u)=\beta(\alpha\mathbf u)\\
(\mathbf u\pm\mathbf v)\pm\mathbf w=\mathbf u\pm(\mathbf v\pm\mathbf w)
$$
> **Commutativity**:
$$
\mathbf u\pm\mathbf v = \mathbf v\pm\mathbf u\\
$$
> **Multiplicative Identity**:
$$
\alpha \times \mathbf{u} = (\alpha u_x;\ \alpha u_y)
$$
> **Distributivity**:
$$
\alpha(\mathbf u\pm\mathbf v)=\alpha u\pm\alpha v\\
(\alpha\pm\beta)\mathbf u=\alpha\mathbf u\pm\beta\mathbf u
$$
>
> **Norm Property**:
$$
\begin{cases}
\|\mathbf u\|\geq0\\
\|\mathbf u\|=0\Leftrightarrow\mathbf u=\mathbf 0\\
\end{cases}\\
\begin{cases}
\|\mathbf u-\mathbf v\|\geq0\\
\|\mathbf u-\mathbf v\|=0\Leftrightarrow\mathbf u=\mathbf v
\end{cases}\\
$$

---
### <center> B.2) Tính chất tích vô hướng (u⋅v) / tích chấm vuông (u⊥v) </center>

:::info
**Dot Product Definition**:
$$
\mathbf u\cdot\mathbf v=u_x\times v_x+u_y\times v_y
$$
**Perp Dot Product Definition**:
$$
\mathbf u^{\perp}\cdot\mathbf v=u_xv_y-u_yv_x
$$
**Perp Dot Relationship**:
$$
perp_{2D}(\mathbf u, \mathbf v) = wedge_{2D}(\mathbf u, \mathbf v) = cross_{2D}(\mathbf u, \mathbf v)\\
$$

`Note:` Chỉ tồn tại ở không gian $2D$
:::
> **Nilpotent Property**:
$$
0 \cdot \mathbf{u} = \mathbf{u} \cdot 0 = 0\\
\mathbf u^{\perp}\cdot\mathbf u^{\perp}=0\\
$$
> **Squared Norm**:
$$
\mathbf u\cdot\mathbf u=\|\mathbf u\|^2
$$
> **Associativity**:
$$
\alpha(\mathbf u\cdot\mathbf v)=(\alpha\mathbf u)\cdot\mathbf v=\mathbf u\cdot(\alpha\mathbf v)\\
\alpha(\mathbf u^{\perp}\cdot\mathbf v)=(\alpha\mathbf u)^{\perp}\cdot\mathbf v=\mathbf u^{\perp}\cdot(\alpha\mathbf v)
$$
> **Basis Vector**:
$$
\widehat{\mathbf i}\cdot\widehat{\mathbf j}=0\\
\widehat{\mathbf i}^{\perp}\cdot\widehat{\mathbf j}=-1\\
$$
> **Commutativity**:
$$
\mathbf u \cdot \mathbf v = \mathbf v \cdot \mathbf u\\
\mathbf u^{\perp} \cdot \mathbf v = \mathbf v \cdot \mathbf u^{\perp}
$$
> **Anticommutativity**:
$$
\mathbf u^{\perp}\cdot\mathbf v=-\mathbf v^{\perp}\cdot\mathbf u\\
$$
> **Multiplicative Distributivity**:
$$
\mathbf u\cdot(\mathbf v\pm\mathbf w)=\mathbf u\cdot\mathbf v\pm\mathbf u\cdot\mathbf w\\
\mathbf u^{\perp}\cdot(\mathbf v\pm\mathbf w)=\mathbf u^{\perp}\cdot\mathbf v\pm\mathbf u^{\perp}\cdot\mathbf w\\
$$
> **Perpendicularity**:
$$
\mathbf u\perp\mathbf v\Leftrightarrow\mathbf u\cdot\mathbf v=0
$$
> **Parallelity**:
$$
\mathbf u\parallel\mathbf v\Leftrightarrow\mathbf u^{\perp}\cdot\mathbf v=0
$$
> **Squared Product**:
$$
(\mathbf u\pm\mathbf v)\cdot(\mathbf u\pm\mathbf v)=(\mathbf u\cdot\mathbf u+\mathbf v\cdot\mathbf v)\pm2\mathbf u\mathbf v\\
(\mathbf u+\mathbf v)\cdot(\mathbf u-\mathbf v)=\mathbf u\cdot\mathbf u-\mathbf v\cdot\mathbf v
$$

---
### <center> B.3) Tính chất tích chéo (u×v) / tích ngoại (u∧v) </center>

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**Cross Product Definition**:
$$
cross_{3D}(\mathbf u, \mathbf v) = \mathbf u\times\mathbf v=
\begin{pmatrix}
u_yv_z - u_zv_y \\
u_zv_x - u_xv_z \\
u_xv_y - u_yv_x \\
\end{pmatrix}\\
cross_{2D}(\mathbf u, \mathbf v) = \|\mathbf u\times\mathbf v\|= (u_xv_y - u_yv_x)\widehat{\mathbf z}
$$
**Wedge Product Definition**:
$$
\mathbf u\wedge\mathbf v=\|\mathbf u\|\|\mathbf v\|\sin(\theta)\widehat{\mathbf n}\\
$$
**Cross Product Relationship**
$$
wedge_{3D}(\mathbf u, \mathbf v) = cross_{3D}(\mathbf u, \mathbf v)\\
perp_{2D}(\mathbf u, \mathbf v) = wedge_{2D}(\mathbf u, \mathbf v) = cross_{2D}(\mathbf u, \mathbf v)\\
$$

**Note:** Ở không gian $2D$ và $3D$ thì tích chéo $\mathbf u\times\mathbf v$ cũng chính là tích ngoại $\mathbf u\wedge\mathbf v$ (nếu coi $e_1, e_2, e_3$ trực chuẩn nhau)

**Note:** Tích chéo không tồn tại ở $1D$ và $4D$ trở lên vì không có khái niệm vector vuông góc duy nhất với mặt phẳng. ($1D$ thì không tồn tại, $4D$ thì không duy nhất). Tích ngoại có tồn tại nhưng ở không gian $1D$ thì khá vô nghĩa.
$$
wedge_{1D}(\mathbf u, \mathbf v) = 0
$$

:::

> **Linearity**:
$$
(\alpha\mathbf u+\beta\mathbf v)\wedge\mathbf w=\alpha(\mathbf u\wedge\mathbf w)+\beta(\mathbf v\wedge\mathbf w)
$$
> **Multiplicative Identity**:
$$
1 \times \mathbf{u} = \mathbf{u} \times 1 = \mathbf{u}\\
$$
> **Nilpotent Property**:
$$
0 \times \mathbf{u} = \mathbf{u} \times 0 = 0\\
\mathbf u\times\mathbf u=0
$$
> **Associativity**:
$$
\alpha(\mathbf u\times\mathbf v)=(\alpha\mathbf u)\times\mathbf v=\mathbf u\times(\alpha\mathbf v)\\
$$
> **Basis Vector**
$$
\widehat{\mathbf i}\times\widehat{\mathbf j}=-1
$$
> **Anticommutativity**:
$$
\mathbf u\times\mathbf v=-\mathbf v\times\mathbf u
$$
> **Multiplicative Distributivity**:
$$
\mathbf u\times(\mathbf v\pm\mathbf w)=(\mathbf u\times\mathbf v)\pm(\mathbf u\times\mathbf w)
$$

---
### <center> B.5) Tính chất tích hình học (uv) và [u,v] </center>

:::info
**Geometric Product Definition**:
$$
\mathbf{uv}=\mathbf u\cdot\mathbf v + \mathbf u\wedge\mathbf v\\
$$
**Commutator Product Definition**:
$$
\mathbf{vu} = [\mathbf u, \mathbf v] =\mathbf u\cdot\mathbf v - \mathbf u\wedge\mathbf v\\
$$

**Note**: Kết quả không phải là scalar hay vector, mà là một cặp scalar và bivector nên ta định nghĩa

$$
\mathbf u\ \&\ \mathbf v = \text{vector2D}\left(\mathbf{uv};\ \mathbf{vu}\right)
$$

:::
> **Symmetric**:
$$
\mathbf u\cdot\mathbf v=\frac{1}{2}(\mathbf u\mathbf v+\mathbf v\mathbf u)
$$
> **Antisymmetric**:
$$
\mathbf u\wedge\mathbf v=\frac{1}{2}(\mathbf u\mathbf v-\mathbf v\mathbf u)
$$
> **Associativity**:
$$
(\mathbf u\mathbf v)\mathbf w=\mathbf u(\mathbf v\mathbf w)
$$
> **Left Distributivity**:
$$
\mathbf u(\mathbf v\pm\mathbf w)=\mathbf u\mathbf v\pm\mathbf u\mathbf w
$$
> **Right Distributivity**:
$$
(\mathbf u\pm\mathbf v)\mathbf w=\mathbf u\mathbf w+\mathbf v\mathbf w
$$
> **Contraction**:
$$
\mathbf u^2=\mathbf u\mathbf u=\|\mathbf u\|^2
$$
> **Inversion**:
$$
\mathbf u^{-1}=\frac{\mathbf u}{\|\mathbf u\|^2}
$$
> **Division**:
$$
\frac{\mathbf u}{\mathbf v} = \frac{\mathbf u\mathbf v}{\|\mathbf v\|^2}
$$
> **Identity**:
$$
(\mathbf u\mathbf v+\mathbf v\mathbf u) = (\mathbf u+\mathbf v)^2-(\mathbf u^2+\mathbf v^2)
$$

---
### <center> B.6) Tính chất giữa các phép nhân </center>

:::info

$$
wedge_{3D}(\mathbf u, \mathbf v) = cross_{3D}(\mathbf u, \mathbf v)\\
perp_{2D}(\mathbf u, \mathbf v) = wedge_{2D}(\mathbf u, \mathbf v) = cross_{2D}(\mathbf u, \mathbf v)\\
$$
:::

> **Unit Vector**:
$$
\underbrace{\|\widehat{\mathbf u}\times\widehat{\mathbf v}\|^2}_{\sin^2\theta}+\underbrace{(\widehat{\mathbf u}\cdot\widehat{\mathbf v})^2}_{\cos^2\theta}=\|\widehat{\mathbf u}\| = \|\widehat{\mathbf v}\| = 1\\
\widehat{\mathbf i}^{\perp}\cdot\widehat{\mathbf j}=\widehat{\mathbf i}\times\widehat{\mathbf j}=\left\|\widehat{\mathbf i}\times\widehat{\mathbf j}\right\|=-1\\
$$
> **Norm Property**:
$$
\begin{cases}
\|\mathbf u\times\mathbf v\|=\mathbf u^{\perp}\cdot\mathbf v=\|\mathbf u\|\|\mathbf v\|\sin\theta \geq 0\\
\|\mathbf u\times\mathbf v\|=0\Leftrightarrow\left[\begin{array}{l}
\mathbf u=\mathbf 0\\
\mathbf v=\mathbf 0\\
\mathbf u\parallel\mathbf v
\end{array}\right.
\end{cases}
$$
> **Jacobi Identity**:
$$
\mathbf u\times(\mathbf v\times\mathbf w)+
\mathbf v\times(\mathbf w\times\mathbf u)+
\mathbf w\times(\mathbf u\times\mathbf v)=
\mathbf 0
$$
> **Associativity/Scalar Triple Product**:
$$
\mathbf u\cdot(\mathbf v\times\mathbf w)=\mathbf w\cdot(\mathbf u\times\mathbf v)=\mathbf v\cdot(\mathbf w\times\mathbf u)
$$
> **Lagrange Identity**:
$$
(\mathbf a\times\mathbf b)\cdot(\mathbf c\times\mathbf d)=(\mathbf a\cdot\mathbf c)(\mathbf b\cdot\mathbf d)-(\mathbf a\cdot\mathbf d)(\mathbf b\cdot\mathbf c)\\
\|\mathbf u^{\perp} \cdot \mathbf v\|^2=\|\mathbf u \times \mathbf v\|^2=\|\mathbf u\|^2\|\mathbf v\|^2-(\mathbf u\cdot\mathbf v)^2\\
$$
> **Product-Product Relation**:
$$
\|\mathbf u\times\mathbf v\|=(\mathbf u^{\perp}\cdot\mathbf v)\widehat{\mathbf z}\\
\mathbf u\cdot\mathbf v=-\mathbf u^{\perp}\cdot\mathbf v^{\perp}\\
$$
> **Product-Angle Relation**:
$$
\frac{\|\mathbf u\times\mathbf v\|}{\mathbf u\cdot\mathbf v}=\frac{\|\mathbf u\|\|\mathbf v\|\sin(\theta)}{\|\mathbf u\|\|\mathbf v\|\cos(\theta)}=\tan(\theta)\\
\frac{\|\mathbf u\times\mathbf v\|}{\|\mathbf u\|\|\mathbf v\|}=\frac{\mathbf u^{\perp}\cdot\mathbf v}{\|\mathbf u\|\|\mathbf v\|}=\sin(\theta)\\
\frac{\mathbf u\cdot\mathbf v}{\|\mathbf u\|\|\mathbf v\|}=\cos(\theta)
$$
> **Cosine Law**:
$$
\begin{cases}
\|\mathbf u\pm\mathbf v\|^2=\|\mathbf u\|^2\pm2\|\mathbf u\|\|\mathbf v\|\cos(\theta)+\|\mathbf v\|^2\\
\theta=\angle(\mathbf u, \mathbf v)\\
\end{cases}
$$
> **Pythagore Identity**:
$$
\mathbf u\cdot\mathbf v=0\Rightarrow\|\mathbf u\pm\mathbf v\|^2=\|\mathbf u\|^2+\|\mathbf v\|^2\\
\|\mathbf u\|^2=(\mathbf u\cdot\widehat{\mathbf v})^2+\|\mathbf u-(\mathbf u\cdot\widehat{\mathbf v})\widehat{\mathbf v}\|^2
$$
> **B.C.S Inequality (Bunhiacopxki-Cauchy-Schwarz)**:
$$
\|\mathbf u\cdot\mathbf v\|\leq\|\mathbf u\|\|\mathbf v\|
$$
> **Triangle Inequality**:
$$
\|\mathbf u+\mathbf v\|\leq\|\mathbf u\|+\|\mathbf v\|
$$
> **Reversed Triangle Inequality**:
$$
\|\mathbf u-\mathbf v\|\geq\|\mathbf u\|-\|\mathbf v\|
$$
> **Extended Triangle Inequality**:
$$
\|\mathbf u-\mathbf w\|\leq\|\mathbf u-\mathbf v\|+\|\mathbf v-\mathbf w\|
$$

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## <center> C. Hàm tiện ích </center>

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### <center> C.1) Hàm tiện ích chuyển đổi hệ </center>
> **Casterian ↦ Polar**: `toPolar(u) =`
$$
\text{pair}(r;\ \theta) = \text{pair}\left(\sqrt{u_x^2+u_y^2};\ \text{atan2}(y,x)\right)=\text{pair}\left(\text{mag}_{\mathbf u};\ \text{polar}_{\mathbf u}\right)
$$
`Note:` pair(r, θ) không nhất thiết phải cùng kiểu dữ liệu
>
> **Polar ↦ Casterian:** `toCasterian(r, θ) =`
$$
\text{vector2D}(u_x, u_y) = \text{vector2D}(r\cos(\theta);\ r\sin(\theta))
$$

---
### <center> C.2) Hàm tiện ích radian </center>

> **Radian ↦ Radian[0, 2π)**: `radNorm(radian θ) =`
$$
((\theta\ \%\ 2\pi)+2\pi)\ \%\ 2\pi
$$
> **Radian ↦ Degree**: `radToDeg(radian θ) =`
==`deg(radian θ) =`==
$$
\frac{\theta}{2\pi} \times 360^\circ
$$
> **Radian ↦ Hour**: `radToHour(radian θ) =`
$$
\frac{\theta}{2\pi} \times 24h
$$
> **Radian ↦ Minute**: `radToMinute(radian θ) =`
$$
\frac{\theta}{2\pi} \times 1440m
$$
> **Radian ↦ Second**: `radToSecond(radian θ) =`
$$
\frac{\theta}{2\pi} \times 86400s
$$
> **Radian ↦ Clock[0h,24h)**: `radToClock(radian θ) =`
$$
\text{vector}(hour, minute, second, nsf)\\
\text{for} \begin{cases}
hour = \left\lfloor\text{toHour}(\text{toNormRad}(\theta))\right\rfloor\ \%\ 24\\
minute = \left\lfloor\text{toMinute}(\text{toNormRad}(\theta))\right\rfloor\ \%\ 60\\
second = \left\lfloor\text{toSecond}(\text{toNormRad}(\theta))\right\rfloor\ \%\ 60\\
nsf = \left\lfloor(\text{toSecond}(\text{toNormRad}(\theta)) - second) \div 10^9\right\rfloor
\end{cases}
$$
`Note:` nsf = nanosecond fraction (but converted to integer)
> **Clock[0h,24h) ↦ Radian[0, 2π)**: `clockToRad(hour, minute, second, nsf) = `
$$
2\pi\times\left(\frac{hour}{24} + \frac{minute}{1440} + \frac{second}{86400} + \frac{nsf}{86400\cdot10^9}\right)
$$
`Default:` $hour:=0, minute:=0, second:=0, nsf:=0$

---
### <center> C.3) Hàm tiện ích degree </center>

> **Degree ↦ Degree[0°, 360°)**: `degNorm(degree ϕ) =`
$$
((\theta\ \%\ 360°)+360°)\ \%\ 360°
$$
> **Degree ↦ Radian**: `degToRad(degree ϕ) =`
==`rad(degree ϕ) =`==
$$
\frac{\phi}{360^\circ}\times 2\pi
$$
> **Degree ↦ Hour**: `degToHour(degree ϕ) =`
$$
\frac{\phi}{360^\circ}\times 24h
$$
> **Degree ↦ Minute**: `degToMinute(degree ϕ) =`
$$
\frac{\phi}{360^\circ}\times 1440m
$$
> **Degree ↦ Second**: `degToSecond(degree ϕ) =`
$$
\frac{\phi}{360^\circ}\times 86400s
$$
> **Degree ↦ Clock[0h,24h)**: `degToClock(radian ϕ)
=`
$$
\text{vector}(hour, minute, second, nanosecond)\\
\text{for}\begin{cases}
hour = \left\lfloor\text{toHour}(\text{toNormDeg}(\phi))\right\rfloor\ \%\ 24\\
minute = \left\lfloor\text{toMinute}(\text{toNormDeg}(\phi))\right\rfloor\ \%\ 60\\
second = \left\lfloor\text{toSecond}(\text{toNormDeg}(\phi))\right\rfloor\ \%\ 60\\
nanosecond = \text{toSecond}(\text{toNormDeg}(\phi)) - second
\end{cases}
$$
> **Clock[0h,24h) ↦ Degree[0°, 360°)**: `clockToRad(hour, minute, second, nsf) = `
$$
360^\circ\times\left(\frac{hour}{24} + \frac{minute}{1440} + \frac{second}{86400} + \frac{nsf}{86400\cdot10^9}\right)
$$
`Default:` $hour:=0, minute:=0, second:=0, nsf:=0$

---
### <center> C.4) Hàm tiện ích số phức </center>

> **Vector ↦ Complex**:
`toComplex(u) = cplx(u) =`
$$
\text{complex}(u_x;\ u_y)
$$
> **Vector * Vector ↦ Complex**:
`complexMultiplication(u, v) = mul(u, v) =`
$$\mathbf u\mathbf v=\text{complex}(u_xv_x-u_yv_y;u_xv_y+u_yv_x)
$$
> **Vector / Vector ↦ Complex**:
`complexDivision(u, v) = div(u, v) =`
$$\mathbf u\div\mathbf v
=\text{complex}\left(\frac{\mathbf u\cdot\mathbf v}{\|\mathbf v\|};\ \frac{\mathbf v\times\mathbf u}{\|\mathbf v\|}\right)
=\text{complex}\left(\frac{u_xv_x + u_yv_y}{u_y^2+v_y^2};\ \frac{u_yv_x-u_xv_y}{u_y^2+v_y^2}\right)
$$

---
### <center> C.5) Hàm tiện ích chỉ hướng </center>

> **Quadrant Function**:
==`quadrantValue(u) =`==
$$
\begin{cases}
-4&,u_x=0\land u_y>0&&\text{phần âm trục tung }Oy^-\\
-3&,u_x<0\land u_y=0&&\text{phần âm trục hoành }Ox^-\\
-2&,u_x=0\land u_y>0&&\text{phần dương trục tung }Oy^+\\
-1&,u_x>0\land u_y=0&&\text{phần dương trục hoành }Ox^+\\
\ \ \ 0&,u_x=0\land u_y=0&&\text{gốc toạ độ } O(0;0)\\
+1&,u_x>0\land u_y>0&&\text{góc phần tư thứ }I\\
+2&,u_x<0\land u_y>0&&\text{góc phần tư thứ }II\\
+3&,u_x<0\land u_y<0&&\text{góc phần tư thứ }III\\
+4&,u_x>0\land u_y<0&&\text{góc phần tư thứ }IV\\
\end{cases}\\
$$
`quadrantString(u) =`
$$
\begin{cases}
\texttt{"Negative Vertical Axis"}&,u_x=0\land u_y>0&&\\
\texttt{"Negative Horizontal Axis"}&,u_x<0\land u_y=0&&\\
\texttt{"Positive Vertical Axis"}&,u_x=0\land u_y>0&&\\
\texttt{"Positive Horizontal Axis"}&,u_x>0\land u_y=0&&\\
\texttt{"Origin Position"}&,u_x=0\land u_y=0&&\\
\texttt{"First Quadrant"}&,u_x>0\land u_y>0&&\\
\texttt{"Second Quadrant"}&,u_x<0\land u_y>0&&\\
\texttt{"Third Quadrant"}&,u_x<0\land u_y<0&&\\
\texttt{"Fourth Quadrant"}&,u_x>0\land u_y<0&&\\
\end{cases}\\
$$
> **Angularity Function**:
==`angularityValue(u, v, [θ]) =`==
$$
\begin{cases}
-3 &,x=y=0&&\mathbf u\text{ có góc bất định với }\mathbf v\\
-2 &,x>y=0&&\mathbf u\text{ tạo thành góc không với }\mathbf v\\
-1 &,xy>0&&\mathbf u\text{ tạo thành góc nhọn với }\\
\ \ \ 0 &,y\neq x=0&&\mathbf u\text{ tạo thành góc vuông với }\mathbf v\\
+1 &,xy<0&&\mathbf u\text{ tạo thành góc tù với }\\
+2 &,x<y=0&&\mathbf u\text{ tạo thàng góc thẳng với }\mathbf v\\
+3 &,xy=\text{NaN}&&\mathbf u\text{ không có góc xác định với }\mathbf v\\
\end{cases}\\
\text{với}\begin{cases}
x = (u_x + v_x)\cos(\theta) - (u_y + v_y)\sin(\theta)\\
y = (u_x + v_x)\sin(\theta) + (u_y + v_y)\cos(\theta)\\
\end{cases}\\
$$
`angularityString(u, v, [θ]) =`
$$
\begin{cases}
\texttt{"Ambiguous Angle"}&,x=y=0&&\mathbf u\text{ có góc bất định với }\mathbf v\\
\texttt{"Zero Angle"}&,x>y=0&&\mathbf u\text{ tạo thành góc không với }\mathbf v\\
\texttt{"Acute Angle"}&,xy>0&&\mathbf u\text{ tạo thành góc nhọn với }\\
\texttt{"Right Angle"}&,y\neq x=0&&\mathbf u\text{ tạo thành góc vuông với }\mathbf v\\
\texttt{"Obtuse Angle"}&,xy<0&&\mathbf u\text{ tạo thành góc tù với }\\
\texttt{"Complete Angle"}&,x<y=0&&\mathbf u\text{ tạo thàng góc thẳng với }\mathbf v\\
\texttt{"Invalid Angle"}&,xy=\text{NaN}&&\mathbf u\text{ không có góc xác định với }\mathbf v\\
\end{cases}\\
\text{với}\begin{cases}
x = (u_x + v_x)\cos(\theta) - (u_y + v_y)\sin(\theta)\\
y = (u_x + v_x)\sin(\theta) + (u_y + v_y)\cos(\theta)\\
\end{cases}\\
$$
`Default:` $\theta := \frac{\pi}{2} = 90^{\circ}$
> **Relative Direction**:
==`relativeValue(u, v, [θ]) =`==
$$
\begin{cases}
-4&,x=0\land y>0&&\mathbf u\text{ vuông góc bên phải với }\mathbf v\\
-3&,x<0\land y=0&&\mathbf u\text{ ngược hướng song song với }\mathbf v\\
-2&,x=0\land y>0&&\mathbf u\text{ vuông góc bên trái với }\mathbf v\\
-1&,x>0\land y=0&&\mathbf u\text{ cùng hướng song song với }\mathbf v\\
\ \ \ 0&,x=0\land y=0&&\mathbf u\text{ trùng với }\mathbf v\\
+1&,x>0\land y>0&&\mathbf u\text{ tạo thành góc nhọn, hướng sang bên trái}\\
+2&,x<0\land y>0&&\mathbf u\text{ tạo thành góc tù, hướng sang bên trái}\\
+3&,x<0\land y<0&&\mathbf u\text{ tạo thành góc tù, hướng sang bên phải}\\
+4&,x>0\land y<0&&\mathbf u\text{ tạo thành góc nhọn, hướng sang bên phải}\\
\end{cases}\\
\text{với}\begin{cases}
x = (u_x + v_x)\cos(\theta) - (u_y + v_y)\sin(\theta)\\
y = (u_x + v_x)\sin(\theta) + (u_y + v_y)\cos(\theta)\\
\end{cases}\\
$$
`relativeString(u, v, [θ]) =`
$$
\begin{cases}
\texttt{"Right-hand Perpendicular"}&,x=0\land y>0\\
\texttt{"Parallel"}&,x<0\land y=0\\
\texttt{"Left-hand Perpendicular"}&,x=0\land y>0\\
\texttt{"Antiparallel"}&,x>0\land y=0\\
\texttt{"Equivalent"}&,x=0\land y=0\\
\texttt{"Left-hand Acute"}&,x>0\land y>0\\
\texttt{"Left-hand Obtuse"}&,x<0\land y>0\\
\texttt{"Right-hand Obtuse"}&,x<0\land y<0\\
\texttt{"Right-hand Acute"}&,x>0\land y<0\\
\end{cases}\\
\text{với}\begin{cases}
x = (u_x + v_x)\cos(\theta) - (u_y + v_y)\sin(\theta)\\
y = (u_x + v_x)\sin(\theta) + (u_y + v_y)\cos(\theta)\\
\end{cases}\\
$$
`Default:` $\theta := 0$
> **Cardinal Direction**:
==`cardinalValue(u) =`==
$$
\begin{cases}
-1 &, (u_x = u_y = 0)\\
0 &, ({1\pi/4} \leq \theta < {3\pi/4}) \\
1 &, ({3\pi/4} \leq \theta < {5\pi/4}) \\
2 &, ({5\pi/4} \leq \theta < {7\pi/4}) \\
3 &, ({7\pi/4} \leq \theta < {8\pi/4}) \lor ({0\pi/4} \le \theta < {1\pi/4}) \\
\end{cases}
$$
`cardinalString(u) =`
$$
\begin{cases}
\texttt{"Center"} &, (u_x = u_y = 0)\\
\texttt{"East"} &, ({7\pi/4} \leq \theta < {8\pi/4}) \lor ({0\pi/4} \le \theta < {1\pi/4}) \\
\texttt{"North"} &, ({1\pi/4} \leq \theta < {3\pi/4}) \\
\texttt{"West"} &, ({3\pi/4} \leq \theta < {5\pi/4}) \\
\texttt{"South"} &, ({5\pi/4} \leq \theta < {7\pi/4}) \\
\end{cases}
$$
> **Intercardinal/Ordinal Direction**:
==`ordinalValue(u) =`==
$$
\begin{cases}
-1 &, (u_x = u_y = 0)\\
0 &, ({15\pi/8} \leq \theta < {16\pi/8}) \lor ({0\pi/8} \le \theta < {1\pi/8}) \\
1 &, ({1\pi/8} \leq \theta < {3\pi/8}) \\
2 &, ({3\pi/8} \leq \theta < {5\pi/8}) \\
3 &, ({5\pi/8} \leq \theta < {7\pi/8}) \\
4 &, ({7\pi/8} \leq \theta < {9\pi/8}) \\
5 &, ({9\pi/8} \leq \theta < {11\pi/8}) \\
6 &, ({11\pi/8} \leq \theta < {13\pi/8}) \\
7 &, ({13\pi/8} \leq \theta < {15\pi/8}) \\
\end{cases}
$$
`ordinalString(u) =`
$$
\begin{cases}
\texttt{"Center"} &, (u_x = u_y = 0)\\
\texttt{"East"} &, ({15\pi/8} \leq \theta < {16\pi/8}) \lor ({0\pi/8} \le \theta < {1\pi/8}) \\
\texttt{"Northeast"} &, ({1\pi/8} \leq \theta < {3\pi/8}) \\
\texttt{"North"} &, ({3\pi/8} \leq \theta < {5\pi/8}) \\
\texttt{"Northwest"} &, ({5\pi/8} \leq \theta < {7\pi/8}) \\
\texttt{"West"} &, ({7\pi/8} \leq \theta < {9\pi/8}) \\
\texttt{"Southwest"} &, ({9\pi/8} \leq \theta < {11\pi/8}) \\
\texttt{"South"} &, ({11\pi/8} \leq \theta < {13\pi/8}) \\
\texttt{"Southeast"} &, ({13\pi/8} \leq \theta < {15\pi/8}) \\
\end{cases}
$$
> **Halfwin Direction**:
==`halfwindValue(u) =`==
$$
\begin{cases}
-1 &, (u_x = u_y = 0)\\
0 &, ({31\pi/16} \leq \theta < {32\pi/16}) \lor ({0\pi/16} \le \theta < {1\pi/16}) \\
1 &, ({1\pi/16} \leq \theta < {3\pi/16}) \\
2 &, ({3\pi/16} \leq \theta < {5\pi/16}) \\
3 &, ({5\pi/16} \leq \theta < {7\pi/16}) \\
4 &, ({7\pi/16} \leq \theta < {9\pi/16}) \\
5 &, ({9\pi/16} \leq \theta < {11\pi/16}) \\
6 &, ({11\pi/16} \leq \theta < {13\pi/16}) \\
7 &, ({13\pi/16} \leq \theta < {15\pi/16}) \\
8 &, ({15\pi/16} \leq \theta < {17\pi/16}) \\
9 &, ({17\pi/16} \leq \theta < {19\pi/16}) \\
10 &, ({19\pi/16} \leq \theta < {21\pi/16}) \\
11 &, ({21\pi/16} \leq \theta < {23\pi/16}) \\
12 &, ({23\pi/16} \leq \theta < {25\pi/16}) \\
13 &, ({25\pi/16} \leq \theta < {27\pi/16}) \\
14 &, ({27\pi/16} \leq \theta < {29\pi/16}) \\
15 &, ({29\pi/16} \leq \theta < {31\pi/16}) \\
\end{cases}
$$
`halfwindString(u) =`
$$
\begin{cases}
\texttt{"Center"} &, (u_x = u_y = 0)\\
\texttt{"East"} &, ({31\pi/16} \leq \theta < {32\pi/16}) \lor ({0\pi/16} \le \theta < {1\pi/16}) \\
\texttt{"Northeast by East"} &, ({1\pi/16} \leq \theta < {3\pi/16}) \\
\texttt{"Northeast"} &, ({3\pi/16} \leq \theta < {5\pi/16}) \\
\texttt{"North by Northeast"} &, ({5\pi/16} \leq \theta < {7\pi/16}) \\
\texttt{"North"} &, ({7\pi/16} \leq \theta < {9\pi/16}) \\
\texttt{"North by Northwest"} &, ({9\pi/16} \leq \theta < {11\pi/16}) \\
\texttt{"Northwest"} &, ({11\pi/16} \leq \theta < {13\pi/16}) \\
\texttt{"Northwest by West"} &, ({13\pi/16} \leq \theta < {15\pi/16}) \\
\texttt{"West"} &, ({15\pi/16} \leq \theta < {17\pi/16}) \\
\texttt{"West by Southwest"} &, ({17\pi/16} \leq \theta < {19\pi/16}) \\
\texttt{"Southwest"} &, ({19\pi/16} \leq \theta < {21\pi/16}) \\
\texttt{"South by Southwest"} &, ({21\pi/16} \leq \theta < {23\pi/16}) \\
\texttt{"South"} &, ({23\pi/16} \leq \theta < {25\pi/16}) \\
\texttt{"South by Southeast"} &, ({25\pi/16} \leq \theta < {27\pi/16}) \\
\texttt{"Southeast"} &, ({27\pi/16} \leq \theta < {29\pi/16}) \\
\texttt{"Southeast by East"} &, ({29\pi/16} \leq \theta < {31\pi/16}) \\
\end{cases}
$$
> **Quarterwin Direction**:
==`quarterwindValue(u) =`==
$$
\begin{cases}
-1 &, (u_x = u_y = 0)\\
0 &, ({63\pi/32} \leq \theta < {64\pi/32}) \lor ({0\pi/32} \le \theta < {1\pi/32}) \\
1 &, ({1\pi/32} \leq \theta < {3\pi/32}) \\
2 &, ({3\pi/32} \leq \theta < {5\pi/32}) \\
3 &, ({5\pi/32} \leq \theta < {7\pi/32}) \\
4 &, ({7\pi/32} \leq \theta < {9\pi/32}) \\
5 &, ({9\pi/32} \leq \theta < {11\pi/32}) \\
6 &, ({11\pi/32} \leq \theta < {13\pi/32}) \\
7 &, ({13\pi/32} \leq \theta < {15\pi/32}) \\
8 &, ({15\pi/32} \leq \theta < {17\pi/32}) \\
9 &, ({17\pi/32} \leq \theta < {19\pi/32}) \\
10 &, ({19\pi/32} \leq \theta < {21\pi/32}) \\
11 &, ({21\pi/32} \leq \theta < {23\pi/32}) \\
12 &, ({23\pi/32} \leq \theta < {25\pi/32}) \\
13 &, ({25\pi/32} \leq \theta < {27\pi/32}) \\
14 &, ({27\pi/32} \leq \theta < {29\pi/32}) \\
15 &, ({29\pi/32} \leq \theta < {31\pi/32}) \\
16 &, ({31\pi/32} \leq \theta < {33\pi/32}) \\
17 &, ({33\pi/32} \leq \theta < {35\pi/32}) \\
18 &, ({35\pi/32} \leq \theta < {37\pi/32}) \\
19 &, ({37\pi/32} \leq \theta < {39\pi/32}) \\
20 &, ({39\pi/32} \leq \theta < {41\pi/32}) \\
21 &, ({41\pi/32} \leq \theta < {43\pi/32}) \\
22 &, ({43\pi/32} \leq \theta < {45\pi/32}) \\
23 &, ({45\pi/32} \leq \theta < {47\pi/32}) \\
24 &, ({47\pi/32} \leq \theta < {49\pi/32}) \\
25 &, ({49\pi/32} \leq \theta < {51\pi/32}) \\
26 &, ({51\pi/32} \leq \theta < {53\pi/32}) \\
27 &, ({53\pi/32} \leq \theta < {55\pi/32}) \\
28 &, ({55\pi/32} \leq \theta < {57\pi/32}) \\
29 &, ({57\pi/32} \leq \theta < {59\pi/32}) \\
30 &, ({59\pi/32} \leq \theta < {61\pi/32}) \\
31 &, ({61\pi/32} \leq \theta < {63\pi/32}) \\
\end{cases}
$$
`quarterwindString(u) =`
$$
\begin{cases}
\texttt{"Center"} &, (u_x = u_y = 0)\\
\texttt{"East"} &, ({63\pi/32} \leq \theta < {64\pi/32}) \lor ({0\pi/32} \le \theta < {1\pi/32}) \\
\texttt{"East by North"} &, ({1\pi/32} \leq \theta < {3\pi/32}) \\
\texttt{"East-Northeast"} &, ({3\pi/32} \leq \theta < {5\pi/32}) \\
\texttt{"Northeast by East"} &, ({5\pi/32} \leq \theta < {7\pi/32}) \\
\texttt{"Northeast"} &, ({7\pi/32} \leq \theta < {9\pi/32}) \\
\texttt{"Northeast by North"} &, ({9\pi/32} \leq \theta < {11\pi/32}) \\
\texttt{"North-Northeast"} &, ({11\pi/32} \leq \theta < {13\pi/32}) \\
\texttt{"North by East"} &, ({13\pi/32} \leq \theta < {15\pi/32}) \\
\texttt{"North"} &, ({15\pi/32} \leq \theta < {17\pi/32}) \\
\texttt{"North by West"} &, ({17\pi/32} \leq \theta < {19\pi/32}) \\
\texttt{"North-Northwest"} &, ({19\pi/32} \leq \theta < {21\pi/32}) \\
\texttt{"Northwest by North"} &, ({21\pi/32} \leq \theta < {23\pi/32}) \\
\texttt{"Northwest"} &, ({23\pi/32} \leq \theta < {25\pi/32}) \\
\texttt{"Northwest by West"} &, ({25\pi/32} \leq \theta < {27\pi/32}) \\
\texttt{"West-Northwest"} &, ({27\pi/32} \leq \theta < {29\pi/32}) \\
\texttt{"West by North"} &, ({29\pi/32} \leq \theta < {31\pi/32}) \\
\texttt{"West"} &, ({31\pi/32} \leq \theta < {33\pi/32}) \\
\texttt{"West by South"} &, ({33\pi/32} \leq \theta < {35\pi/32}) \\
\texttt{"West-Southwest"} &, ({35\pi/32} \leq \theta < {37\pi/32}) \\
\texttt{"Southwest by West"} &, ({37\pi/32} \leq \theta < {39\pi/32}) \\
\texttt{"Southwest"} &, ({39\pi/32} \leq \theta < {41\pi/32}) \\
\texttt{"Southwest by South"} &, ({41\pi/32} \leq \theta < {43\pi/32}) \\
\texttt{"South-Southwest"} &, ({43\pi/32} \leq \theta < {45\pi/32}) \\
\texttt{"South by West"} &, ({45\pi/32} \leq \theta < {47\pi/32}) \\
\texttt{"South"} &, ({47\pi/32} \leq \theta < {49\pi/32}) \\
\texttt{"South by East"} &, ({49\pi/32} \leq \theta < {51\pi/32}) \\
\texttt{"South-Southeast"} &, ({51\pi/32} \leq \theta < {53\pi/32}) \\
\texttt{"Southeast by South"} &, ({53\pi/32} \leq \theta < {55\pi/32}) \\
\texttt{"Southeast"} &, ({55\pi/32} \leq \theta < {57\pi/32}) \\
\texttt{"Southeast by East"} &, ({57\pi/32} \leq \theta < {59\pi/32}) \\
\texttt{"East-Southeast"} &, ({59\pi/32} \leq \theta < {61\pi/32}) \\
\texttt{"East by South"} &, ({61\pi/32} \leq \theta < {63\pi/32}) \\
\end{cases}
$$

---
### <center> C.6) Biểu tượng </center>

`quadrantSymbol(u) =`
$$
\begin{cases}
\texttt{'⍖'}&,u_x=0\land u_y>0&&\text{phần âm trục tung }Oy^-\\
\texttt{'⍅'}&,u_x<0\land u_y=0&&\text{phần âm trục hoành }Ox^-\\
\texttt{'⍏'}&,u_x=0\land u_y>0&&\text{phần dương trục tung }Oy^+\\
\texttt{'⍆'}&,u_x>0\land u_y=0&&\text{phần dương trục hoành }Ox^+\\
\texttt{'⨁'}&,u_x=0\land u_y=0&&\text{gốc toạ độ } O(0;0)\\
\texttt{'Ⅰ'}&,u_x>0\land u_y>0&&\text{góc phần tư thứ }I\\
\texttt{'Ⅱ'}&,u_x<0\land u_y>0&&\text{góc phần tư thứ }II\\
\texttt{'Ⅲ'}&,u_x<0\land u_y<0&&\text{góc phần tư thứ }III\\
\texttt{'Ⅳ'}&,u_x>0\land u_y<0&&\text{góc phần tư thứ }IV\\
\end{cases}\\
$$
`angularitySymbol(u, v, [θ]) =`
$$
\begin{cases}
\texttt{'𐽗'}&,x=y=0&&\mathbf u\text{ có góc bất định với }\mathbf v\\
\texttt{'⥉'}&,x>y=0&&\mathbf u\text{ tạo thành góc không với }\mathbf v\\
\texttt{'⦟'}&,xy>0&&\mathbf u\text{ tạo thành góc nhọn với }\\
\texttt{'⦜'}&,y\neq x=0&&\mathbf u\text{ tạo thành góc vuông với }\mathbf v\\
\texttt{'⦦'}&,xy<0&&\mathbf u\text{ tạo thành góc tù với }\\
\texttt{'⥈'}&,x<y=0&&\mathbf u\text{ tạo thàng góc thẳng với }\mathbf v\\
\texttt{'∅'}&,xy=\text{NaN}&&\mathbf u\text{ không có góc xác định với }\mathbf v\\
\end{cases}\\
\text{với}\begin{cases}
x = (u_x + v_x)\cos(\theta) - (u_y + v_y)\sin(\theta)\\
y = (u_x + v_x)\sin(\theta) + (u_y + v_y)\cos(\theta)\\
\end{cases}\\
$$
`relativeSymbol(u, v, [θ]) =`
$$
\begin{cases}
\texttt{'⍆'}&,x=0\land y>0&&\mathbf u\text{ vuông góc bên phải với }\mathbf v\\
\texttt{'⇅'}&,x<0\land y=0&&\mathbf u\text{ ngược hướng song song với }\mathbf v\\
\texttt{'⍅'}&,x=0\land y>0&&\mathbf u\text{ vuông góc bên trái với }\mathbf v\\
\texttt{'⇈'}&,x>0\land y=0&&\mathbf u\text{ cùng hướng song song với }\mathbf v\\
\texttt{'≡'}&,x=0\land y=0&&\mathbf u\text{ trùng với }\mathbf v\\
\texttt{'⦭'}&,x>0\land y>0&&\mathbf u\text{ tạo thành góc nhọn, hướng sang bên trái}\\
\texttt{'⦯'}&,x<0\land y>0&&\mathbf u\text{ tạo thành góc tù, hướng sang bên trái}\\
\texttt{'⦮'}&,x<0\land y<0&&\mathbf u\text{ tạo thành góc tù, hướng sang bên phải}\\
\texttt{'⦬'}&,x>0\land y<0&&\mathbf u\text{ tạo thành góc nhọn, hướng sang bên phải}\\
\end{cases}\\
\text{với}\begin{cases}
x = (u_x + v_x)\cos(\theta) - (u_y + v_y)\sin(\theta)\\
y = (u_x + v_x)\sin(\theta) + (u_y + v_y)\cos(\theta)\\
\end{cases}\\
$$
`cardinalSymbol(u) =`
$$
\begin{cases}
\texttt{'◆'} &, (u_x = u_y = 0)\\
\texttt{'⮙'} &, ({1\pi/4} \leq \theta < {3\pi/4}) \\
\texttt{'⮘'} &, ({3\pi/4} \leq \theta < {5\pi/4}) \\
\texttt{'⮛'} &, ({5\pi/4} \leq \theta < {7\pi/4}) \\
\texttt{'⮚'} &, ({7\pi/4} \leq \theta < {8\pi/4}) \lor ({0\pi/4} \le \theta < {1\pi/4}) \\
\end{cases}
$$
`ordinalSymbol(u) =`
$$
\begin{cases}
\texttt{'◆'} &, u_x = u_y = 0\\
\texttt{'⮚'} &, ({15\pi/8} \leq \theta < {16\pi/8}) \lor ({0\pi/8} \le \theta < {1\pi/8}) \\
\texttt{'⬈'} &, ({1\pi/8} \leq \theta < {3\pi/8}) \\
\texttt{'⮙'} &, ({3\pi/8} \leq \theta < {5\pi/8}) \\
\texttt{'⬉'} &, ({5\pi/8} \leq \theta < {7\pi/8}) \\
\texttt{'⮘'} &, ({7\pi/8} \leq \theta < {9\pi/8}) \\
\texttt{'⬋'} &, ({9\pi/8} \leq \theta < {11\pi/8}) \\
\texttt{'⮛'} &, ({11\pi/8} \leq \theta < {13\pi/8}) \\
\texttt{'⬊'} &, ({13\pi/8} \leq \theta < {15\pi/8}) \\
\end{cases}
$$
`halfwindSymbol(u) =`
$$
\begin{cases}
\texttt{'◆'} &, (u_x = u_y = 0)\\
\texttt{'⮚'} &, ({31\pi/16} \leq \theta < {32\pi/16}) \lor ({0\pi/16} \le \theta < {1\pi/16}) \\
\texttt{'⮫'} &, ({1\pi/16} \leq \theta < {3\pi/16}) \\
\texttt{'⬈'} &, ({3\pi/16} \leq \theta < {5\pi/16}) \\
\texttt{'⮭'} &, ({5\pi/16} \leq \theta < {7\pi/16}) \\
\texttt{'⮙'} &, ({7\pi/16} \leq \theta < {9\pi/16}) \\
\texttt{'⮬'} &, ({9\pi/16} \leq \theta < {11\pi/16}) \\
\texttt{'⬉'} &, ({11\pi/16} \leq \theta < {13\pi/16}) \\
\texttt{'⮪'} &, ({13\pi/16} \leq \theta < {15\pi/16}) \\
\texttt{'⮘'} &, ({15\pi/16} \leq \theta < {17\pi/16}) \\
\texttt{'⮨'} &, ({17\pi/16} \leq \theta < {19\pi/16}) \\
\texttt{'⬋'} &, ({19\pi/16} \leq \theta < {21\pi/16}) \\
\texttt{'⮮'} &, ({21\pi/16} \leq \theta < {23\pi/16}) \\
\texttt{'⮛'} &, ({23\pi/16} \leq \theta < {25\pi/16}) \\
\texttt{'⮯'} &, ({25\pi/16} \leq \theta < {27\pi/16}) \\
\texttt{'⬊'} &, ({27\pi/16} \leq \theta < {29\pi/16}) \\
\texttt{'⮩'} &, ({29\pi/16} \leq \theta < {31\pi/16}) \\
\end{cases}
$$
`quarterwindSymbol(u) =`
$$
\begin{cases}
\texttt{"◆"} &, (u_x = u_y = 0)\\
\texttt{"⮚"} &, ({63\pi/32} \leq \theta < {64\pi/32}) \lor ({0\pi/32} \le \theta < {1\pi/32}) \\
\texttt{"⮫-"} &, ({1\pi/32} \leq \theta < {3\pi/32}) \\
\texttt{"⮫"} &, ({3\pi/32} \leq \theta < {5\pi/32}) \\
\texttt{"⮫+"} &, ({5\pi/32} \leq \theta < {7\pi/32}) \\
\texttt{"⬈"} &, ({7\pi/32} \leq \theta < {9\pi/32}) \\
\texttt{"⮭-"} &, ({9\pi/32} \leq \theta < {11\pi/32}) \\
\texttt{"⮭"} &, ({11\pi/32} \leq \theta < {13\pi/32}) \\
\texttt{"⮭+"} &, ({13\pi/32} \leq \theta < {15\pi/32}) \\
\texttt{"⮙"} &, ({15\pi/32} \leq \theta < {17\pi/32}) \\
\texttt{"⮬-"} &, ({17\pi/32} \leq \theta < {19\pi/32}) \\
\texttt{"⮬"} &, ({19\pi/32} \leq \theta < {21\pi/32}) \\
\texttt{"⮬+"} &, ({21\pi/32} \leq \theta < {23\pi/32}) \\
\texttt{"⬉"} &, ({23\pi/32} \leq \theta < {25\pi/32}) \\
\texttt{"⮪-"} &, ({25\pi/32} \leq \theta < {27\pi/32}) \\
\texttt{"⮪"} &, ({27\pi/32} \leq \theta < {29\pi/32}) \\
\texttt{"⮪+"} &, ({29\pi/32} \leq \theta < {31\pi/32}) \\
\texttt{"⮘"} &, ({31\pi/32} \leq \theta < {33\pi/32}) \\
\texttt{"⮨-"} &, ({33\pi/32} \leq \theta < {35\pi/32}) \\
\texttt{"⮨"} &, ({35\pi/32} \leq \theta < {37\pi/32}) \\
\texttt{"⮨+"} &, ({37\pi/32} \leq \theta < {39\pi/32}) \\
\texttt{"⬋"} &, ({39\pi/32} \leq \theta < {41\pi/32}) \\
\texttt{"⮮-"} &, ({41\pi/32} \leq \theta < {43\pi/32}) \\
\texttt{"⮮"} &, ({43\pi/32} \leq \theta < {45\pi/32}) \\
\texttt{"⮮+"} &, ({45\pi/32} \leq \theta < {47\pi/32}) \\
\texttt{"⮛"} &, ({47\pi/32} \leq \theta < {49\pi/32}) \\
\texttt{"⮯-"} &, ({49\pi/32} \leq \theta < {51\pi/32}) \\
\texttt{"⮯"} &, ({51\pi/32} \leq \theta < {53\pi/32}) \\
\texttt{"⮯+"} &, ({53\pi/32} \leq \theta < {55\pi/32}) \\
\texttt{"⬊"} &, ({55\pi/32} \leq \theta < {57\pi/32}) \\
\texttt{"⮩-"} &, ({57\pi/32} \leq \theta < {59\pi/32}) \\
\texttt{"⮩"} &, ({59\pi/32} \leq \theta < {61\pi/32}) \\
\texttt{"⮩+"} &, ({61\pi/32} \leq \theta < {63\pi/32}) \\
\end{cases}
$$

---
---
---
---
---
## <center> D. Hàm tính toán </center>

---
### <center> D.1) Hàm tính giá trị thông dụng </center>

> **Zero Check**: `u.isZero() = true`
==`⇔ is0(u) = true`==
$$
\Leftrightarrow \mathbf u = \mathbf 0
\Leftrightarrow\begin{cases}
u_x = 0\\
u_y = 0
\end{cases}
$$
>
> **Horizontal Position**: `u.horizontal() =`
==`u.x() =`==
$$
u_x
$$
>
> **Vertical Position**: `u.vertical() =`
==`u.y() =`==
$$
u_y
$$
>
> **Norm:** `u.norm(p) =`
==`norm(u, p) =`==
$$
\|\mathbf u\|_p = (|u_x|^p+|u_y|^p)^{(1/p)}
$$
`Default:` $p := 2$
>
> **Length/Abs**: `u.length() = u.abs() =`
==`len(u) = abs(u) =`==
$$
\|\mathbf u\|=\sqrt{u_x^2+u_y^2}
$$
> **Squared Length**: `u.squaredLength() =`
==`sq(u)`==
$$
u_x^2+u_y^2
$$
`Note:` Thường dùng với số nguyên $\in \mathbb{Z}$ cho bớt sai số
> **Magnitude Position**: `u.r() = u.magnitude()`
==`r(u) = mag(u) =`==
$$
\text{mag}_{\mathbf u}=\|\mathbf u\|
$$
> **Polar Angle**: `u.theta() = u.polarAngle() = polarAngle(u) =`
==`theta(u) = polar(u) =`==
$$
\text{atan2}(u_y; u_x)
$$
>
> **Angle Between Vectors**: `u.angle(v) =`
==`angle(u, v) =`==
$$
\theta = (\text{atan2}(\mathbf u\times\mathbf v,\ \mathbf u\cdot\mathbf v) + \pi)\ \%\ 2\pi
$$
> **Interior Angle**: `u.interiorAngle(v) =`
==`interior(u, v) =`==
$$
\theta_{int} = 2\pi - \theta_{ext} = \cos^{-1}\left(\frac{u\cdot v}{\| u \| \| v \|}\right)
$$
> **Exterior Angle**: `u.exteriorAngle(v) =`
==`exterior(u, v) =`==
$$
\theta_{ext} = 2\pi - \theta_{int} = \sin^{-1}\left(\frac{u\times v}{\| u \| \| v \|}\right)
$$
> **Manhattan/Taxicab Distance**: `manhattanDistance(u, v) =`
==`u.taxicab(v) =`==
$$
\| \mathbf u - \mathbf v\|_x = \| \mathbf v - \mathbf u \|_x = \|u_x - v_x\| + \|u_y - v_y\|
$$
> **Straight/Euclidean Distance**: `euclideanDistance(u, v)`
==`u.dist(v) =`==
$$
\| \mathbf u - \mathbf v\| = \| \mathbf v - \mathbf u\| = \sqrt{(u_x-v_x)^2 + (u_y-v_y)^2}
$$
> **Squared Euclidean Distance**: `squaredDistance(u, v) =`
==`u.sqdist(v)`==
$$
\| \mathbf u - \mathbf v\|^2 = \| \mathbf v - \mathbf u\|^2 = (u_x-v_x)^2 + (u_y-v_y)^2
$$
`Note:` Không dùng công thức $\text{dist}(u, v)^2$ để tránh sai số

---
### <center> D.2) Hàm tính giá trị ít dùng </center>
> **Scalar Projection**: `u.scalarProj(θ) = u.scalarProj(v) =`
$$
\|\mathbf u\|\cos\theta=\frac{\mathbf u\cdot\mathbf v}{\|\mathbf v\|}=\frac{u_xv_x+u_yv_y}{\sqrt{v_x^2+v_y^2}}
$$
> **Scalar Rejection**: `u.scalarRej(θ) = u.scalarRej(v) =`
$$
\|\mathbf u\|\sin\theta=\frac{\mathbf u\cdot\mathbf v^{\perp}}{\|\mathbf v\|}=\frac{u_yv_x-u_xv_y}{\sqrt{v_x^2+v_y^2}}
$$

---
### <center> D.3) Hàm tính vector </center>

> **Vector Projection**: `u.vectorProjection(v) =`
==`proj(u, v) =`==
$$
\text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{\| \mathbf v \|^2} \times \mathbf v = \widehat{\mathbf u}\mathbf u.\text{scalarProj}(\mathbf v)
$$
>
> **Vector Rejection**: `u.vectorRejection(v) =`
==`rej(u, v) =`==
$$
\text{rej}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u}^{\perp}\cdot \mathbf{v}}{\| \mathbf v \|^2} \times \mathbf v = \mathbf u-\text{proj}_\mathbf v(\mathbf u)
$$
>
> **Dot Product**: `u.dotProduct(v) =`
==`dot(u, v) =`==
$$
\mathbf u\cdot\mathbf v=u_x\times v_x+u_y\times v_y
$$
> **Perp Dot Product**: `u.perpProduct(v) =`
==`perp(u, v) =`==
$$
\mathbf u^{\perp}\cdot\mathbf v=u_x\times v_y-u_y\times v_x
$$
> **2D Cross Product**: `u.crossProduct(v) =`
==`cross(u, v) =`==
$$
\mathbf u\times\mathbf v=u_x\times v_y-u_y\times v_x
$$
> **Wedge Product**: `u.wedgeProduct(v)`
==`wedge(u, v) =`==
$$
\mathbf u\wedge\mathbf v=u_x\times v_y-u_y\times v_x
$$
**Geometric & Commutator Product**: `u.geoProduct(v)`
==`geomul(u, v) =`==
$$
\text{vector2D}(\mathbf{uv};\ \mathbf{vu})
$$
**Vector Geometric Inverse**: `u.geoInverse()`
==`geoinv(u) =`==
$$
\mathbf u^{-1} = \frac{\mathbf u}{\|\mathbf u\|^2} = \frac{\widehat{\mathbf u}}{\|\mathbf u\|}
$$
**Vector Geometric Division**: `u.geoDivision(v)`
==`geodiv(u, v) =`==
$$
\frac{\mathbf u}{\mathbf v} = \mathbf u\left(\mathbf v^{-1}\right) = \frac{\mathbf u\mathbf v}{\|\mathbf v\|^2}
$$
**Vector Cross Division**: `crossDivision(u×v, v, θ) =`
$$
\mathbf u = \frac{\|\mathbf u\times\mathbf v\|\cot(\theta)\mathbf v-\mathbf v\times(\mathbf u\times \mathbf v)}{\|\mathbf v\|^2}
$$

---
### <center> D.4) Hàm biến đổi vector </center>

#### <center> D.4.a) Hàm biến đổi tuyến tính vector </center>

> **Swap Transformation**: `u.swap(v)`
> **Negation Transformation**: `u.negation() =`
$$
-\mathbf{u} = \text{vector2D}(-u_x;\ -u_y)
$$
> **Conjugation Transformation**: `u.conjugation() =`
$$
\overline{\mathbf u} = \text{vector2D}(u_x;\ -u_y)
$$
>
> **Translation Transformation**: `u.translate(v) = u.translate(vx, vy) =`
$$
\mathbf u \mapsto \mathbf u + \mathbf v = \text{vector2D}(u_x + v_x;\ u_y + v_y)
$$
> **Reversion Transformation**: `u.reverse() =`
$$
\text{rev}_{\mathbf u} = \text{vector2D}(u_y;\ u_x)
$$
> **Reflection Transformation**: `u.reflection(v) =`
$$
\overset{\leftrightarrow\hspace{-6.5pt}{\small|}}{\text{ Ref}}_{\mathbf{v}}(\mathbf{u}) = 2\text{proj}_\mathbf v(\mathbf u) - \mathbf u = 2\frac{\mathbf u\cdot\mathbf v}{\|\mathbf v\|^2}\mathbf v-\mathbf u
$$
> **Mirror-X/Flip-Y Transformation**: `u.mirror_x() = u.flip_y() =`
$$
\overset{\leftarrow\hspace{-5.5pt}{\small|}}{\text{ R}}(\mathbf u) = \text{vector2D}(u_x;\ -u_y)
$$
> **Mirror-Y/Flip-X Transformation**: `u.mirror_y() = u.flip_x() =`
$$
\overset{\downarrow\hspace{-6.5pt}{\overline{\ \ \ }}}{\text{ R}}(\mathbf u)=\text{vector2D}(-u_x;\ u_y)
$$
> **Mirror-XY/Flip-XY/Negation Transformation**: `u.mirror_xy() = u.flip_xy() = u.negation() =`
$$
\overset{\swarrow\hspace{-6.5pt}{\small\backslash}}{\text{ R}}(\mathbf u)=\text{vector2D}(-u_x;\ -u_y)
$$
> **Scale Transformation**: `u.scale(α, [β]) =`
$$
\overset{\nearrow\hspace{-9.5pt}\swarrow}{\text{Scale}}_\text{ }(\mathbf u, \alpha, \beta) =
\text{vector2D}(\alpha u_x;\ \beta u_y)
$$
`Default:` $\beta := \alpha$
`Note:`
> > Khi $\left[\begin{array}{l}\alpha = 1, \beta > 1\\\alpha > 1, \beta = 1\end{array}\right.$ thì gọi là hàm `stretch` ($\nwarrow\hspace{-9pt}\nearrow$)
> > Khi $\left[\begin{array}{l}\alpha < 1, \beta = 1\\\alpha = 1, \beta < 1\end{array}\right.$ thì gọi là hàm `squash` ($\swarrow\hspace{-9pt}\searrow$)
> > Khi $\left[\begin{array}{l}\alpha < 1, \beta > 1\\\alpha > 1, \beta < 1\end{array}\right.$ thì gọi là hàm `squeeze` ($\uparrow\downarrow$)

#### <center> D.4.c) Hàm biến đổi khác </center>

> **Unitization Transformation**: `u.unit() = u.normalized() =`
$$
\widehat{\mathbf u} = \text{vector2D}\left(\frac{u_x}{\|\mathbf u\|}, \frac{u_y}{\|\mathbf u\|}\right)
$$
`Note:` Cẩn thận khi $\mathbf u=\mathbf 0$, và cẩn thận nhầm với hàm`norm(u, p)`
> **Reciprocity Transformation**: `u.reciprocal() =`
$$
\text{rec}_{\mathbf u} = \text{vector2D}\left(\frac{1}{u_x};\ \frac{1}{u_y}\right)
$$
`Note:` Cẩn thận khi $u_x=0$ hoặc $u_y = 0$
> **Rotation Transformation**: `u.rotate(θ, [isClockwise]) =`
$$
\begin{cases}
\overset{\curvearrowleft}{R}_{\theta}(u)=\left(u_x\cos(\theta)-u_y\sin(\theta);\ u_y\cos(\theta)+u_x\sin(\theta)\right)&,isClockwise=false\\
\overset{\curvearrowright}{R}_{\theta}(u)=\left(u_x\cos(\theta)+u_y\sin(\theta);\ u_y\cos(\theta)-u_x\sin(\theta)\right)&,isClockwise=true\\
\end{cases}
$$
`Default:` $\theta := \frac{\pi}{2}$ (quay $90^\circ$) và $isClockwise := true$ (quay cùng chiều kim đồng hồ)
>
> **Clamp Transformation**: `u.clamp(α, [β]) = `
$$
\text{vector2D}\left(\text{sign}(u_x)\min(|u.x|, |\alpha|) ;\ \text{sign}(u_y)\min(|u.y|, |\beta|)\right)
$$
`Default:` $\beta := \alpha$
> **Shear Transformation**: `u.shear(α, [β]) =`
$$
\overset{\Lsh\Rsh}{\text{T}}_\text{shear}(\mathbf u, \alpha,\beta)=\text{vector2D}\left(u_x+\alpha u_y;\ u_y+\beta u_x\right)
$$
`Default:` $\beta := 0$
> **Linear Transformation**: `u.shear(α1, α2, β1, β2) =`
$$
\text{T}(\mathbf u, \alpha_1, \alpha_2, \beta_1, \beta_2)=\text{vector2D}\left(\alpha_1u_x+\alpha_2u_y;\ \beta_1u_y+\beta_2u_x\right)
$$
> **Component-wise Transformation**: `u.componentwise(f) =`
$$
\mathbf u\odot f=\text{vector2D}\left(f(u_x);\ f(u_y)\right)
$$

---
---
---
---
---
## <center> E. Hàm kiểm tra </center>

---
### <center> E.1) Kiểm tra tính bằng nhau </center>

> **Sign Function**:
==`sign(scalar x) =`==
$$
[x > 0] - [x < 0]
=\begin{cases}
+1&,x>0\\
\ \ 0&,x=0\\
-1&,x<0\\
\end{cases}
$$
`Note:` $0$ và $x$ nên cùng kiểu dữ liệu, và dấu $[x = y]$ là [iverson brackets](https://en.wikipedia.org/wiki/Iverson_bracket)
`Note:` Có thể xem `quadrant()` là sign function cho vector
>
> **Float Equality**: `isEqual(scalar u, scalar v, high-precision ε) = true`
$$
\Leftrightarrow \begin{cases}
|u_x-v_x| \leq \varepsilon\\
|u_y-v_y| \leq \varepsilon\\
\end{cases}
$$
`Default:` $\varepsilon = 10^{-9}$
> **Equality Function**:
==`equalityValue(scalar x, scalar y) =`==
$$
\begin{cases}
-9&, \vert x-y\vert\leq10^{-9}\\
-8&, 10^{-9} < \vert x-y\vert\leq10^{-8}\\
-7&, 10^{-8} < \vert x-y\vert\leq10^{-7}\\
-6&, 10^{-7} < \vert x-y\vert\leq10^{-6}\\
-5&, 10^{-6} < \vert x-y\vert\leq10^{-5}\\
-4&, 10^{-5} < \vert x-y\vert\leq10^{-4}\\
-3&, 10^{-4} < \vert x-y\vert\leq10^{-3}\\
-2&, 10^{-3} < \vert x-y\vert\leq10^{-2}\\
-1&, 10^{-2} < \vert x-y\vert\leq10^{-1}\\
\ \ \ 0&,x\not\approx y\\
\end{cases}
$$

---
### <center> E.2) Kiểm tra tính trực giao </center>

> **Orthogonal Check**: `isOrthogonal(u, v) = true`
$$
\Leftrightarrow \left[\begin{array}{l}
\mathbf u\cdot\mathbf v=0\\
\mathbf u\times\mathbf v=0\\
\end{array}\right.
$$
> **Dilation Factor**: `u.dilation(v) =`
$$
\begin{cases}
\frac{u_x}{v_x}&,(v_xv_y\neq 0) \land \left(\frac{u_x}{v_x} = \frac{u_y}{v_y}\in\mathbb{R}\right)\\
+\infty&,\left(\frac{u_x}{v_x}=\frac{u_y}{v_y}=+\infty\right)\\
-\infty&,\left(\frac{u_x}{v_x}=\frac{u_y}{v_y}=-\infty\right)\\
\text{NaN}&,\left(\frac{u_y}{v_y}=\text{NaN}\right)\lor\left(\frac{u_x}{v_x}=\text{NaN}\right)\lor\left(\frac{u_y}{v_y}=\text{NaN}\right)\lor\left(\frac{u_x}{v_x} \neq \frac{u_y}{v_y}\right)\\
\end{cases}
$$
> **Collinear Check**: `isColinear(u, v) = true`
$$
\Leftrightarrow \exists k=\text{dilation}(\mathbf u, \mathbf v)\to\mathbf u=k\mathbf v\\
\Leftrightarrow\mathbf u\times\mathbf v=0
$$
> **Parallel Check**: `isParallel(u, v) = true`
$$
\Leftrightarrow \begin{cases}
\mathbf u\times\mathbf v=0\\
\mathbf u\cdot\mathbf v>0
\end{cases}
$$
> **Antiparallel Check**: `isAntiparallel(u, v) = true`
$$
\Leftrightarrow \begin{cases}
\mathbf u\times\mathbf v=0\\
\mathbf u\cdot\mathbf v<0
\end{cases}
$$
> **Zero Angle Check**: `isZeroAngle(u, v) = true`
$$
\Leftrightarrow \measuredangle(\mathbf{u}, \mathbf{v})=0^\circ
\Leftrightarrow\theta=0
\Leftrightarrow\mathbf u\cdot\mathbf v=\|\mathbf u\|\cdot\|\mathbf v\|
$$
> **Right Angle Check**: `isPerpendicular(u, v) = isRightAngle(u, v) = true`
$$
\Leftrightarrow \measuredangle(\mathbf{u},\mathbf{v})=90^\circ
\Leftrightarrow\theta=\frac{\pi}{2}
\Leftrightarrow\mathbf u\perp\mathbf v
\Leftrightarrow\mathbf u\cdot\mathbf v=0
$$
> **Straight Angle Check**: `isStraightAngle(u, v) = true`
$$
\Leftrightarrow \measuredangle(\mathbf{u}, \mathbf{v})=180^\circ
\Leftrightarrow\theta=\pi
\Leftrightarrow\mathbf u\cdot\mathbf v=-\|\mathbf u\|\cdot\|\mathbf v\|
$$
>

---
### <center> E.3) Kiểm tra góc xiên </center>

> **Left-Right Side Indicator**: `sideX(u, [v]) =`
$$
\begin{cases}
+1&,\mathbf u\cdot\mathbf v>0&&\text{same side}\\
\ \ \ 0&,\mathbf u\cdot\mathbf v=0&&\text{perpendicular}\\
-1&,\mathbf u\cdot\mathbf v<0&&\text{opposite side}\\
\end{cases}
$$
`Default:` $\mathbf v = \mathbf 0 \Leftrightarrow side(\mathbf u) = sign(u_y)$
`Note:` so với đường thẳng vuông góc với phương của $\mathbf u$
> **Top-Down Side Indicator**: `sideY(u, [v]) =`
$$
\begin{cases}
+1&,\mathbf u\times\mathbf v>0&&\text{same half}\\
\ \ \ 0&,\mathbf u\times\mathbf v=0&&\text{collinear}\\
-1&,\mathbf u\times\mathbf v<0&&\text{opposite half}\\
\end{cases}
$$
`Default:` $\mathbf v = \mathbf 0$
`Note:` so với đường thẳng đi qua $\mathbf u$
>
> **Acute Angle Check**: `isAcuteAngle(u, v) = true`
$$
\Leftrightarrow \measuredangle(\mathbf{u},\mathbf{v})\in(0^\circ,
90^\circ)
\Leftrightarrow 0 < \theta < \frac{\pi}{2}
\Leftrightarrow\mathbf u\cdot\mathbf v>0
$$
> **Obtuse Angle Check**: `isObtuseAngle(u, v) = true`
$$
\Leftrightarrow \measuredangle(\mathbf{u},\mathbf{v})\in(90^\circ,180^\circ)
\Leftrightarrow \frac{\pi}{2} < \theta < \pi
\Leftrightarrow\mathbf u\cdot\mathbf v<0
$$
> **Reflex Angle Check**: `isReflexAngle(u, v) = !isReflexAngle(v, u) = true`
$$
\Leftrightarrow \measuredangle(\mathbf{u},\mathbf{v})\in(-180^\circ,0^\circ)
\Leftrightarrow \pi < \theta < 2\pi
\Leftrightarrow\mathbf u\times\mathbf v<0
$$

---
### <center> E.4) Hàm kiểm tra hướng góc </center>
> **Orientability Function**: `orient(u, v) =`
$$
\begin{cases}
+1&,\mathbf u\times\mathbf v>0&&\text{counter-clockwise}\\
\ \ \ 0&,\mathbf u\times\mathbf v=0&&\text{collinear}\\
-1&,\mathbf u\times\mathbf v<0&&\text{clockwise}\\
\end{cases}
$$
> **Polar Order**: `polarOrder(u, v, [o]) =`
$$
\begin{cases}
-1&,LHS<RHS\\
\ \ \ 0&,LHS=RHS\\
+1&,LHS>RHS\\
\end{cases}\\
\text{for }\begin{cases}
LHS = tuple(sideY(u-w)&, 0&, sq(u-w))\\
RHS = tuple(sideY(v-w)&, cross(u-w, v-w)&, sq(v-w))\\
\end{cases}
$$
`Default:` $\mathbf w = \mathbf 0$
> **Counter-Clockwise Direction**: `isCounterClockwise(u, v) = true`
`ccw(u,v)`
$$
\Leftrightarrow \measuredangle(\mathbf{u},\mathbf{v})\in(0^\circ,180^\circ)
\Leftrightarrow\mathbf u\times\mathbf v>0
$$
> **Collinear Direction**: `isColinear(u, v) = true`
`cln(u,v)`
$$
\Leftrightarrow \exists k\in\mathbb{R},\mathbf u=k\mathbf v
\Leftrightarrow\mathbf u\times\mathbf v=0
$$
> **Clockwise Direction**: `isClockwise(u,v) = isReflexAngle(u, v) = true`
`cw(u,v)`
$$
\Leftrightarrow \measuredangle(\mathbf{u},\mathbf{v})\in(180^\circ,360^\circ)
\Leftrightarrow\mathbf u\times\mathbf v<0
$$
> **Unoriented Direction**: `isUnoriented(u, v) = true`
$$
\Leftrightarrow \mathbf u=\mathbf v=\mathbf 0
$$

---
---
---
---
---
## <center> F. Cài đặt </center>

---
### <center> F.1) Nhập xuất khởi tạo </center>

template<class scalar>
class vector2D {
    scalar x, y;
    ...
};
using Point = vector2D<int>

> **Default Initialization**:
> `Constructor `==`vector2D()`==
> ```cpp
> : x(0), y(0) {}
> ```
> **Initialize from uniform scalar**:
> `Constructor `==`vector2D(scalar x, scalar y)`==
> ```cpp
> : x(x), y(y) {}
> ```
> **Initialize from non-uniform scalar**:
> `Constructor `==`vector2D(scalar x, scalar y)`==

template<typename LHS, typename RHS, typename T = typename conditional
	<is_integral<LHS>::value&&is_integral<RHS>::value,long long,double>::type>
vector2D<T> makeVector2D(const LHS& x, const RHS& y) {
    return vector2D<T>(static_cast<T>(x), static_cast<T>(y));
}

> **Initialize from vector**:
> `Assignment Operator`==`vector = other_vector`==
> ```cpp
> { x = other.x; y = other.y; }
> ```
> **Memory Free**:
> `Destructor `==`~vector2D()`==
> ```cpp
> { delete x; delete y; }
> ```
> **Input (cin)**:
`Insertion Operator`==`>> vector`==`↦ &istream cin`
> ```cpp
> { return cin >> x >> y, cin; }
> ```
> **Output (cout, cerr)**:
`Extraction Operator`==`<< vector`==`↦ &ostream cout`
> ```cpp
> { return cout << "(" << x << "; " << y << ")", cout; }
> ```
> `Note:` Nếu muốn tạo luồng cerr và cout khác nhau thì
> ```cpp
> if (cout.rdbuf() == cerr.rdbuf()) { /// cerr
> cout << "(" << u.x << "; " << u.y << ")";
> } else { /// cout
> cout << u.x << " " << u.y;
> }
> return cout;
> ```

---
### <center> F.2) Toán tử so sánh </center>

---
#### <center> F.2.a) Toán tử kiểm tra vector 0 </center>
> **Non-zero Check**:
> `Explixit Bool: `==`(vector)`==`↦ boolean`
> ```cpp
> { return !isZero(); }
> ```
> **Zero Check**:
> `Exclamation Operator: `==`!vector`==`↦ boolean`
> ```cpp
> { return isZero(); }
> ```

---
#### <center> F.2.b) So sánh vector theo góc quay </center>
> **Less Than Operator**:
> `Less Than Operator: `==`vector < other_vector`==`↦ bool`
> ```cpp
> { return polarOrder(vector, other_vector); }
> ```
> **Equal To Operator**:
> `Equal To Operator: `==`vector <= other_vector`==`↦ bool`
> ```cpp
> { return isEqual(vector.x, other_vector.x)
> && isEqual(vector.y, other_vector.y); }
> ```
> **Less Than or Equal To Operator**:
> `Less Than or Equal To Operator: `==`vector <= other_vector`==`↦ bool`
> ```cpp
> { return (u < v) || (u == v); }
> ```
> **Not Equal To**:
> `Not Equal To Operator: `==`vector >= other_vector`==`↦ bool`
> ```cpp
> { return !(u = v); }
> ```
> **Greater Than or Equal To Operator**:
> `Greater Than or Equal To Operator: `==`vector >= other_vector`==`↦ bool`
> ```cpp
> { return !(u < v); }
> ```
> **Greater Than Operator**:
> `Greater Than Operator: `==`vector > other_vector`==`↦ bool`
> ```cpp
> { return !(u <= v); }
> ```

---
### <center> F.3) Toán tử biến đổi vector </center>

---
#### <center> F.3.a) Symmetry Reflection </center>
> **Vector Itself**:
> `Unary Plus Operator:`==`+vector`==`↦ vector`
> ```cpp
> { return *this; }
> ```
> **Vector Symmetry Reflection**:
> `Unary Minus Operator:`==`-vector`==`↦ vector`
> ```cpp
> { return negation(); }
> ```
> **Vector Coordinate Swap**:
> `Logical NOT Operator`==`~vector`==`↦ vector`
> ```cpp
> { return reversion(); }
> ```

---
#### <center> F.3.b) Translating </center>
> **Vector Sumation**:
> `Addition Operator:`==`vector + vector`==`↦ vector`
> ```cpp
> { return translate(+vector); }
> ```
> **Vector Difference**:
> `Subtraction Operator:`==`vector - vector`==`↦ vector`
> ```cpp
> { return translate(-vector); }
> ```
> **Translating**:
> `Addition Assignment Operator:`==`vector += vector`==`↦ &vector`
> ```cpp
> { return *this = translate(+vector); }
> ```
> **Reverse Translating**:
> `Subtraction Assignment Operator:`==`vector -= vector`==`↦ &vector`
> ```cpp
> { return *this = translate(-vector); }
> ```
> **Vector Horizontal Slide**:
> `Pre-increment Operator: `==`++vector`==`↦ &vector`
> ```cpp
> { return *this = translate(+1, 0); }
> ```
> **Vector Horizontal Reverse Slide**:
> `Pre-decrement Operator: `==`--vector`==`↦ &vector`
> ```cpp
> { return *this = translate(-1, 0); }
> ```
> **Vector Vertical Slide**:
> `Post-increment Operator: `==`vector++`==`↦ &vector`
> ```cpp
> { return *this = translate(0, +1); }
> ```
> **Vector Vertical Reverse Slide**:
> `Post-decrement Operator: `==`vector--`==`↦ &vector`
> ```cpp
> { return *this = translate(0, -1); }
> ```
`Note:` `Post-* Operator` có thể trả về `&vector` https://ideone.com/HRgB56

---
#### <center> F.3.c) Scaling </center>

> **Stretched Vector**:
> `LHS Multiplication Operator:`==`vector * scalar`==`↦ vector`
> `RHS Multiplication Operator:`==`scalar * vector`==`↦ vector`
> ```cpp
> { return scale(scalar); }
> ```
> **Stretching**:
> `Multiplication Assignment Operator: `==`vector *= scalar`==`↦ &vector`
> ```cpp
> { return *this = scale(scalar); }
> ```
> **Squashed Vector**:
> `LHS Division Operator: `==`vector / scalar`==`↦ vector`
> ```cpp
> { return scale(1 / scalar); }
> ```
> **Reciprocal Squashed Vector**:
> `RHS Division Operator: `==`scalar / vector`==`↦ vector`
> ```cpp
> { return reciprocal().scale(scalar); }
> ```
> `Note:` Trùng với vector division nếu xem vector như một multivector
> **Squashing**:
> `Division Assignment Operator: `==`vector /= scalar`==`↦ &vector`
> ```cpp
> { return *this = scale(1 / scalar); }
> ```

---
#### <center> F.3.d) Wrapping </center>

> **Clamped Vector**:
> `Logical AND Operator`==`vector & other_vector`==`↦ vector`
> ```cpp
> { return clamp(other_vector.x, other_vector.y); }
> ```
> **Clamping**:
> `Logical AND Assignment Operator`==`vector &= other_vector`==`↦ vector`
> ```cpp
> { return *this = clamp(other_vector.x, other_vector.y); }
> ```
> **Wrapped Vector**:
> `LHS Modulo Operator: `==`vector % scalar`==`↦ vector`
> ```cpp
> { return vector2D(scalar % x, scalar % y); }
> ```
> **Reversed Wrapped Vector**:
> `RHS Modulus Operator: `==`scalar % vector`==`↦ vector`
> ```cpp
> { return vector2D(x % scalar, y % scalar); }
> ```
> **Wrapping**:
> `Modulo Assignment Operator: `==`vector %= scalar`==`↦ &vector`
> ```cpp
> { return *this = vector2D(x % scalar, y % scalar); }
> ```

---
### <center> F.4) Toán tử số học với vector </center>

> **Vector Horizontal Position**:
> `Overloading Operator`==`vector[0]`==`↦ &scalar`
> ```cpp
> { return vector.x; }
> ```
> **Vector Vertical Position**:
> `Overloading Operator`==`vector[1]`==`↦ &scalar`
> ```cpp
> { return vector.y; }
> ```
> **Dot Product/Inner Product**:
> `Multiplication Operator`==`vector * other_vector`==`↦ scalar`
> `Logical OR Operator`==`vector | other_vector`==`↦ scalar`
> ```cpp
> { return dot(*this, other_vector); }
> ```
> **Perp Dot Product/2D Wedge Product/2D Cross Product**:
> `Remainder Operator`==`vector % other_vector`==`↦ scalar`
> `Logical XOR Operator`==`vector ^ other_vector`==`↦ scalar`
> ```cpp
> { return perp(*this, other_vector); }
> ```
> **Vector Dilation Ratio**:
> `Division Operator`==`vector / other_vector`==`↦ scalar`
> ```cpp
> { return dilation(*this, other_vector); }
> ```
> **Vector Distance**:
> `Shift Right Operator`==`vector >> other_vector`==`↦ scalar`
> ```cpp
> { return euclideanDistance(*this, other_vector); }
> ```
> **Vector Squared Distance**:
> `Shift Left Operator`==`vector << other_vector`==`↦ scalar`
> ```cpp
> { return squaredDistance(*this, other_vector); }
> ```

---
## <center> Z. Footnote </center>
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