Section02 Lengths and Dot Products

Dot Product

Defination

$$\boldsymbol{v} \cdot \boldsymbol{w} = \sum_{i=1}^{n} (v_{i}w_{i}) = \Vert \boldsymbol{v} \Vert \cdot \Vert \boldsymbol{w} \Vert \cdot cos(\angle_{\boldsymbol{v},\boldsymbol{w}})$$

Properties

• when the result of dot product is zero, the two vectors are perpendicular.
• $\boldsymbol{v} \cdot \boldsymbol{w} = \boldsymbol{w} \cdot \boldsymbol{v}$

Application

• Business $$\begin{array}{c} \left\{ \begin{array}{ll} \text{price vector} & (p_{1},p_{2}, \cdots, p_{n})\\ \text{quantities buy and sell} & (q_{1}, q_{2}, \cdots, q_{n}) \end{array} \right.\\\\ \text{Income} = (p_{1},p_{2}, \cdots, p_{n}) \cdot (q_{1}, q_{2}, \cdots, q_{n}) = \sum_{i=1}^{n}p_{i}q_{i} \end{array}$$

Length and Unit Vectors

Length

$$\text{length of } \boldsymbol{v} \text{ squared: }\Vert \boldsymbol{v} \Vert^{2} = \boldsymbol{v} \cdot \boldsymbol{v}$$

Unit Vectors

$$\begin{array}{c} \textbf{A unit vector } \boldsymbol{u} \textbf{ is a vector whose length equals one.} \\\\ \boldsymbol{u} \cdot \boldsymbol{u} = 1 \\\\ \end{array}$$

Properties

• $(\text{cos}\theta, \text{sin}\theta)$ is a unit vector. (极坐标)
• $\frac{\boldsymbol{v}}{\Vert \boldsymbol{v} \Vert}$ is the unit vector in the same direction as $\boldsymbol{v}$.

The Angle Between Two Vectors

Proof: when $\boldsymbol{v} \cdot \boldsymbol{w} = 0$, $\boldsymbol{v}$ and $\boldsymbol{w}$ are perpendicular. $$\begin{array}{ll} \because & \boldsymbol{v} \perp \boldsymbol{w}\\ \therefore & \Vert \boldsymbol{v} - \boldsymbol{w} \Vert^{2} = \Vert \boldsymbol{v} \Vert^{2} + \Vert \boldsymbol{w} \Vert^{2} \\ \therefore & \sum_{i=1}^{n} (v_{i} - w_{i})^{2} = \sum_{i=1}^{n}v_{i}^{2} + \sum_{i=1}^{n}w_{i}^{2} \\ \therefore & \sum_{i=1}^{n}(v_{i}^{2} - 2v_{i}w_{i} + w_{i}^{2}) = \sum_{i=1}^{n}v_{i}^{2} + \sum_{i=1}^{n}w_{i}^{2} \\ \therefore & 2\sum_{i=1}^{n}v_{i}w_{i} = 0 \\ \therefore & \sum_{i=1}^{n}v_{i}w_{i} = \boldsymbol{v} \cdot \boldsymbol{w} = 0\\ \therefore & \text{if } \boldsymbol{v} \cdot \boldsymbol{w} = 0,\text{ then } \boldsymbol{v} \perp \boldsymbol{w} \\ \end{array}$$

Proof: If $\text{slope}_{\boldsymbol{v}} \cdot \text{slope}_{\boldsymbol{w}} = -1 \text{ (} \boldsymbol{v} \text{ and } \boldsymbol{w} \text{ are two-dimession vectors.)}$, then $\boldsymbol{v} \perp \boldsymbol{w}$ $$\begin{array}{ll} \because & \text{slope}_{\boldsymbol{v}} = \frac{v2}{v1} \text{ and slope}_{\boldsymbol{w}} = \frac{w2}{w1} \\ \because & \text{slope}_{\boldsymbol{v}} \cdot \text{slope}_{\boldsymbol{w}} = -1 \\ \therefore & \frac{v_{2}w_{2}}{v_{1}w_{1}} = -1 \\ & v_{2}w_{2} + v_{1}w_{1} = 0 \\ & \boldsymbol{v} \cdot \boldsymbol{w} = 0 \\ \therefore & \boldsymbol{v} \perp \boldsymbol{w} \\ \end{array}$$

From dot product to angle

• When the result of dot product is positive, the angle between two vectors is less than 90 degrees.
• When the result of dot product is negative, the angle between two vectors is greater than 90 degrees.
• The borderline is the zero. where the angle is 90 degree.

Calculating the cosine of angle between two vectors

• Unit Vectors $\boldsymbol{u}$ and $\boldsymbol{U}$ at angle $\theta$ have $\boldsymbol{u} \cdot \boldsymbol{U} = cos \theta$. Certainly $\vert \boldsymbol{u} \cdot \boldsymbol{U} \vert \le 1$
  • 当两个单位向量点乘时，点乘的结果为两个单位向量夹角的余弦值。
• If $\boldsymbol{v}$ and $\boldsymbol{w}$ are nonzero vectors, then $\frac{\boldsymbol{v} \cdot \boldsymbol{w}}{\Vert \boldsymbol{v} \Vert \Vert \boldsymbol{w} \Vert} = \text{cos} \theta$

Properties

• Schwarz Inequality $$\vert \boldsymbol{v} \cdot \boldsymbol{w} \vert \le \Vert \boldsymbol{v} \Vert \cdot \Vert \boldsymbol{w} \Vert$$

  • proof $$\begin{array}{ll} \because & \frac{\boldsymbol{v} \cdot \boldsymbol{w}}{\Vert \boldsymbol{v} \Vert \cdot \Vert \boldsymbol{w} \Vert} = \text{cos } \theta \le1 \\ \because & \Vert \boldsymbol{v} \Vert \cdot \Vert \boldsymbol{w} \Vert > 0\\ \therefore & \boldsymbol{v} \cdot \boldsymbol{w} \le \Vert \boldsymbol{v} \Vert \cdot \Vert \boldsymbol{w} \Vert \\ \end{array}$$

• Triangle Inequality $$\Vert \boldsymbol{v} + \boldsymbol{w} \Vert \le \Vert \boldsymbol{v} \Vert + \Vert \boldsymbol{w} \Vert$$

  • proof

    根据定义，三角形两边之和大于第三边

Computing In Programing

• $\boldsymbol{v} \cdot \boldsymbol{w}$

  # This is a Julia Example
  v = [1,2,3];
  w = [4,5,6];
  
  # v cdot w 
  v' * w;
  

• Angle Computing

  # This is a Julia Example
  using LinearAlgebra
  v = [1,2,3];
  w = [4,5,6];
  
  cosine = (v' * w) / (norm(v) * norm(w))
  theta = acos(cosine)