🔰vector

LaTeX ⟩ vector

\|\vec{u}\|    % vector magnitude

💈基本

example	code
$(a_1, a_2, \cdots, a_n)$	👉 向量 % vector (a_1, a_2, \cdots, a_n)
$\|\vec{u}\|$	👉 向量長度 % vector magnitude / norm / length \|\vec{u}\|
$\mathbf{u}\times\mathbf{v}$	👉 外積 % cross product \mathbf{u}\times\mathbf{v}
$\mathbf{u}\cdot\mathbf{v}$	👉 內積 % dot product \mathbf{u}\cdot\mathbf{v}
$(\mathbf{u}\times\mathbf{v})\times\mathbf{w}$	👉 向量三重積 % vector triple product (\mathbf{u}\times\mathbf{v})\times\mathbf{w}
$(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}$	👉 純量三重積 % scalar triple produc (\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}
$\begin{bmatrix} x & y & z \end{bmatrix}$	% row vector \begin{bmatrix} x & y & z \end{bmatrix}
$\begin{bmatrix} x \\ y \\ z \end{bmatrix}$	% column vector \begin{bmatrix} x \\ y \\ z \end{bmatrix}
$\begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$	% dot product as matrix multiplication \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}
$\{ \mathbf{v_1}, \mathbf{v_2}, \cdots, \mathbf{v_n} \}$	% set of vectors \{ \mathbf{v_1}, \mathbf{v_2}, \cdots, \mathbf{v_n} \}

💈大型

example	code
$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$	👉 向量長度 % vector length \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}
$\|\mathbf{v}\|^2 = v_1^2 + v_2^2 + \cdots + v_n^2$	% vector length \|\mathbf{v}\|^2 = v_1^2 + v_2^2 + \cdots + v_n^2
$a_1\mathbf{v_1} + a_2 \mathbf{v_2} \cdots + a_n \mathbf{v_n}$	👉 線性組合 % linear combination a_1\mathbf{v_1} + a_2 \mathbf{v_2} \cdots + a_n \mathbf{v_n}
$\text{proj}_{\mathbf{v}} ( \textcolor{red}{\mathbf{u}} ) = \left( \dfrac {\textcolor{red}{\mathbf{u}}\cdot\mathbf{v}} {\mathbf{v}\cdot\mathbf{v}} \right) \mathbf{v}$	👉 向量的垂直分解 % projection of u onto v \text{proj}_{\mathbf{v}} ( \textcolor{red}{\mathbf{u}} ) = \left( \dfrac {\textcolor{red}{\mathbf{u}}\cdot\mathbf{v}} {\mathbf{v}\cdot\mathbf{v}} \right) \mathbf{v}
$\text{perp}_{\mathbf{v}}( \textcolor{red}{\mathbf{u}} ) = \textcolor{red}{\mathbf{u}} - \left( \dfrac{ \textcolor{red}{\mathbf{u}}\cdot\mathbf{v} }{ \mathbf{v}\cdot\mathbf{v} } \right) \mathbf{v}$	👉 向量的垂直分解 % perpendicular vector \text{perp}_{\mathbf{v}}( \textcolor{red}{\mathbf{u}} ) = \textcolor{red}{\mathbf{u}} - \left( \dfrac{ \textcolor{red}{\mathbf{u}}\cdot\mathbf{v} }{ \mathbf{v}\cdot\mathbf{v} } \right) \mathbf{v}
$\left( \begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix} , \begin{vmatrix} u_3 & u_1 \\ v_3 & v_1 \end{vmatrix} , \begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix} \right)$	👉 空間外積 % cross product \left( \begin{vmatrix} u_2 & u_3 \\ v_2 & v_3 \end{vmatrix} , \begin{vmatrix} u_3 & u_1 \\ v_3 & v_1 \end{vmatrix} , \begin{vmatrix} u_1 & u_2 \\ v_1 & v_2 \end{vmatrix} \right)

⬇️ 應用

• 內積 ⟩ 餘弦定理

• 射影向量的矩陣表示法

📘 手冊

• KaTeX > Accents