点乘 Dot (scalar) Product

$$ \vec{a} \cdot \vec{b}=\|\vec{a}\|\|\vec{b}\| \cos \theta $$

$$ \cos \theta=\frac{\vec{a} \cdot \vec{b}}{\|\vec{a}\|\|\vec{b}\|} $$

对于单位向量，模都是 1，因此可以化简成：

$$ \cos \theta=\hat{a} \cdot \hat{b} $$

符合交换律、结合律、分配律

$$ \vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a} \\ $$

$$ \vec{a} \cdot(\vec{b}+\vec{c})=\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c} $$

$$ (k \vec{a}) \cdot \vec{b}=\vec{a} \cdot(k \vec{b})=k(\vec{a} \cdot \vec{b}) $$

Dot Product in Cartesian Coordinates 笛卡尔坐标系

Component-wise multiplication, then adding up

In 2D

$$ \vec{a} \cdot \vec{b}=\left(\begin{array}{l}x_{a} \\y_{a}\end{array}\right) \cdot\left(\begin{array}{l}x_{b} \\y_{b}\end{array}\right)=x_{a} x_{b}+y_{a} y_{b} $$

In 3D

$$ \vec{a} \cdot \vec{b}=\left(\begin{array}{c}x_{a} \\y_{a} \\z_{a}\end{array}\right) \cdot\left(\begin{array}{c}x_{b} \\y_{b} \\z_{b}\end{array}\right)=x_{a} x_{b}+y_{a} y_{b}+z_{a} z_{b} $$

Dot Product in Graphics

• Find angle between two vectors (e.g. cosine of angle between light source and surface)
• Finding projection of one vector on another