# 內積

[線性代數](/math/linear.md) ⟩ [向量](/math/linear/vec.md) ⟩ [運算](/math/linear/vec/op.md) ⟩ 內積  (:u6307: 同義詞："<mark style="color:purple;">**dot product**</mark>", "<mark style="color:purple;">**inner product**</mark>", "<mark style="color:purple;">**scalar produc**</mark>t")

{% hint style="warning" %}
⭐️ 這裡定義的<mark style="color:purple;">**內積**</mark><mark style="color:yellow;">**只適用於**</mark> $$\mathbb{R}ⁿ$$ ，更廣義的內積，要依相對的[向量空間](/math/linear/space.md)而定❗️&#x20;
{% endhint %}

{% hint style="success" %}
若 $$\mathbf{u} = ( u\_1 , u\_2 , \cdots , u\_n )$$、 $$\mathbf{v} = ( v\_1 , v\_2 , \cdots , v\_n )$$ ，定義向量<mark style="color:purple;">**內積**</mark>為：

$$\mathbf{u}\cdot \mathbf{v}=u\_1v\_1 + u\_2v\_2 + \cdots + u\_nv\_n$$
{% endhint %}

{% tabs %}
{% tab title="🔸 性質" %}
{% hint style="success" %}

1. (<mark style="color:yellow;">**交換律**</mark>)：$$\mathbf{u}\cdot\mathbf{v} = \mathbf{v}\cdot\mathbf{u}$$&#x20;
2. (<mark style="color:yellow;">**分配律**</mark>)：$$\mathbf{u}\cdot (\mathbf{v} + \mathbf{w}) = \mathbf{u}\cdot\mathbf{v} + \mathbf{u}\cdot\mathbf{w}$$&#x20;
3. (<mark style="color:red;">**類**</mark><mark style="color:yellow;">**結合律**</mark>)：$$k(\mathbf{u}\cdot\mathbf{v}) = (k\mathbf{u})\cdot\mathbf{v} = \mathbf{u}\cdot(k\mathbf{v})$$
   {% endhint %}
   {% endtab %}

{% tab title="🔴 主題" %}

* [內積的矩陣表示法](/math/linear/vec/op/dot/in-matrix.md)
* [內積的餘弦定理](/math/linear/vec/op/dot/cos-rule.md)
  {% endtab %}

{% tab title="⭐️ 重點" %}
{% hint style="info" %}
若用[轉置矩陣](/math/linear/matrix/op/transpose.md)與[矩陣乘法](/math/linear/matrix/op/mult.md)，則<mark style="color:purple;">**內積**</mark>也可以寫成：

$$\mathbf{u}\cdot \mathbf{v} = \mathbf{u}^T \mathbf{v}$$    （註：用[矩陣乘法](/math/linear/matrix/op/mult.md)計算<mark style="color:purple;">**內積**</mark>時，[向量](/math/linear/vec.md)通常用「<mark style="color:yellow;">**行向量**</mark>」表示）
{% endhint %}
{% endtab %}

{% tab title="⬇️ 應用" %}

* [向量的垂直分解](/math/linear/vec/decomp/perp.md) (證明垂直分解是<mark style="color:yellow;">**兩個線性變換**</mark>)
  {% endtab %}

{% tab title="🖥️ 影片" %}
{% embed url="<https://youtu.be/51-V4nD1Z80?t=153>" %}
{% endtab %}

{% tab title="👥 相關" %}

* [向量長度](/math/linear/vec/norm.md)可以轉為<mark style="color:purple;">**內積**</mark>。
* [矩陣乘法](/math/linear/matrix/op/mult.md)的定義與<mark style="color:purple;">**內積**</mark>有關。
* 比較： [行 ⨉ 列](/math/linear/matrix/op/mult/outer-product.md) ("outer product")、[外積](/math/linear/vec/op/cross.md)
* [複數乘法](/math/num/complex/mult.md)
* [四元數內積](/math/num/quaternion/op/dot.md)就是[向量](/math/linear/vec.md)<mark style="color:purple;">**內積**</mark>。
* [R³ 中的旋轉](/math/linear/space/transform/rotate3d.md)
  {% endtab %}

{% tab title="📗 參考" %}

* [ ] Mathematics for 3D Game Programming & Computer Graphics (2nd Edition, 2004) &#x20;
  {% endtab %}
  {% endtabs %}


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