> 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/linear/vec/op/triple/vec.md).

# 向量三重積

🚧 under construction

[線性代數](/math/linear.md) ⟩ [向量](/math/linear/vec.md) ⟩ [運算](/math/linear/vec/op.md) ⟩ [三重積](/math/linear/vec/op/triple.md) ⟩ 向量三重積

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( :star: <mark style="color:red;">**只**</mark>適用於<mark style="color:yellow;">**座標空間**</mark>:exclamation:)

$$(\mathbf{u}\times\mathbf{v})\times\mathbf{w}  = \begin{vmatrix}   \mathbf{u}\cdot\mathbf{w} & \mathbf{v}\cdot\mathbf{w} \   \mathbf{u} & \mathbf{v} \ \end{vmatrix}$$

:u6307: 同義詞："<mark style="color:purple;">**vector triple product**</mark>"
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{% tab title="🔸 性質" %}
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1. $$(\mathbf{u}\times\mathbf{v})\times\mathbf{w} \ne \mathbf{u}\times (\mathbf{v}\times\mathbf{w})$$  (:star:「[空間外積](/math/linear/vec/op/cross/3d.md)」<mark style="color:red;">**沒有**</mark>「<mark style="color:yellow;">**結合律**</mark>」:exclamation:)
2. $$(\mathbf{u}\times\mathbf{v})\times\mathbf{u} = \mathbf{u}\times(\mathbf{v}\times\mathbf{u})$$
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{% tab title="👥 相關" %}

* 外積 ⟩ [性質](/math/linear/vec/op/cross.md#xing-zhi)&#x20;
* [TeX](/math/tool/tex.md) ⟩  [vector](/math/tool/tex/vec.md)
  {% endtab %}

{% tab title="📘 手冊" %}

* wiki ⟩ [triple product](https://en.wikipedia.org/wiki/Triple_product)
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{% tab title="🎖 證明" %}
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$$(\mathbf{u}\times\mathbf{v})\times\mathbf{u} = \mathbf{u}\times(\mathbf{v}\times\mathbf{u})$$
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🎖 證明：

$$(\mathbf{u}\times\mathbf{v})\times\mathbf{u} = -(\mathbf{v}\times\mathbf{u})\times\mathbf{u} = \mathbf{u}\times(\mathbf{v}\times\mathbf{u})$$ ▨
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