> For the complete documentation index, see [llms.txt](https://lochiwei.gitbook.io/math/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://lochiwei.gitbook.io/math/tool/tex/vec.md). # vector [LaTeX](/math/tool/tex.md) โŸฉ vector {% hint style="success" %} ```tex \|\vec{u}\| % vector magnitude ``` {% endhint %} {% tabs %} {% tab title="๐Ÿ’ˆๅŸบๆœฌ" %}
examplecode
(a_1, a_2, \cdots, a_n)

๐Ÿ‘‰ ๅ‘้‡

% vector
(a_1, a_2, \cdots, a_n)
\|\vec{u}\|

๐Ÿ‘‰ ๅ‘้‡้•ทๅบฆ

% vector magnitude / norm / length
\|\vec{u}\|
\mathbf{u}\times\mathbf{v}

๐Ÿ‘‰ ๅค–็ฉ

% cross product
\mathbf{u}\times\mathbf{v}
\mathbf{u}\cdot\mathbf{v}

๐Ÿ‘‰ ๅ…ง็ฉ

% dot product
\mathbf{u}\cdot\mathbf{v}
(\mathbf{u}\times\mathbf{v})\times\mathbf{w}

๐Ÿ‘‰ ๅ‘้‡ไธ‰้‡็ฉ

% vector triple product
(\mathbf{u}\times\mathbf{v})\times\mathbf{w}
(\mathbf{u}\times\mathbf{v})\cdot\mathbf{w}

๐Ÿ‘‰ ็ด”้‡ไธ‰้‡็ฉ

% 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} \}
{% endtab %} {% tab title="๐Ÿ’ˆๅคงๅž‹" %}
examplecode
\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

๐Ÿ‘‰ ๅ‘้‡้•ทๅบฆ

% 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}

๐Ÿ‘‰ ็ทšๆ€ง็ต„ๅˆ

% 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}

๐Ÿ‘‰ ๅ‘้‡็š„ๅž‚็›ดๅˆ†่งฃ

% 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}

๐Ÿ‘‰ ๅ‘้‡็š„ๅž‚็›ดๅˆ†่งฃ

% 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)

๐Ÿ‘‰ ็ฉบ้–“ๅค–็ฉ

% 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)
{% endtab %} {% tab title="โฌ‡๏ธ ๆ‡‰็”จ" %} * [ๅ…ง็ฉ](/math/linear/vec/op/dot.md) โŸฉ [้ค˜ๅผฆๅฎš็†](/math/linear/vec/op/dot/cos-rule.md) * [ๅฐ„ๅฝฑๅ‘้‡็š„็Ÿฉ้™ฃ่กจ็คบๆณ•](/math/linear/vec/proj/matrix.md) {% endtab %} {% tab title="๐Ÿ“˜ ๆ‰‹ๅ†Š" %} * KaTeX > [Accents](https://katex.org/docs/supported.html#accents) {% endtab %} {% endtabs %}