vector | 數學

example

code

$(a_1, a_2, \cdots, a_n)$

% 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}

$% dot product as matrix multiplication \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$ $% dot product as matrix multiplication \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$ $\begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$

% 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} \}

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)

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