使用Python进行矢量投影

矢量是具有大小（即长度）和方向的几何对象。向量一般用一条连接起点A和终点B的具有一定方向的线段来表示，如下图所示，记为$$\overrightarrow{AB}$$

一个向量在另一个向量上的投影

向量的投影$$\overrightarrow{u}$$到另一个向量上$$\overrightarrow{v}$$给出为
$$proj_{\vec{v}}({\vec{u}}) = \frac{\vec{u}.\vec{v}}{||\vec{v}||^{2}} \vec{v}$$

在Python中计算向量投影到另一个向量上：

# import numpy to perform operations on vector
import numpy as np
  
u = np.array([1, 2, 3])   # vector u
v = np.array([5, 6, 2])   # vector v:
  
# Task: Project vector u on vector v
  
# finding norm of the vector v
v_norm = np.sqrt(sum(v**2))    
  
# Apply the formula as mentioned above
# for projecting a vector onto another vector
# find dot product using np.dot()
proj_of_u_on_v = (np.dot(u, v)/v_norm**2)*v
  
print("Projection of Vector u on Vector v is: ", proj_of_u_on_v)

输出：

Projection of Vector u on Vector v is:  [1.76923077 2.12307692 0.70769231]

一个用于将一个向量投影到另一个向量上的线性代码：

(np.dot(u, v)/np.dot(v, v))*v

将向量投影到平面上

向量的投影$$\overrightarrow{u}$$在一个平面上的计算是通过减去的分量$$\overrightarrow{u}$$它与平面正交$$\overrightarrow{u}$$ .
$$proj_{Plane}({\vec{u}}) ={\vec{u}} - proj_{\vec{n}}({\vec{u}}) = {\vec{u}} - \frac{\vec{u}.\vec{n}}{||\vec{n}||^{2}} \vec{n}$$
在哪里， $$\overrightarrow{n}$$是平面法向量。

在Python中计算向量投影到平面上：

# import numpy to perform operations on vector
import numpy as np
  
# vector u 
u = np.array([2, 5, 8])       
  
# vector n: n is orthogonal vector to Plane P
n = np.array([1, 1, 7])       
   
# Task: Project vector u on Plane P
  
# finding norm of the vector n 
n_norm = np.sqrt(sum(n**2))    
   
# Apply the formula as mentioned above
# for projecting a vector onto the orthogonal vector n
# find dot product using np.dot()
proj_of_u_on_n = (np.dot(u, n)/n_norm**2)*n
  
# subtract proj_of_u_on_n from u: 
# this is the projection of u on Plane P
print("Projection of Vector u on Plane P is: ", u - proj_of_u_on_n)

输出：

Projection of Vector u on Plane P is:  [ 0.76470588  3.76470588 -0.64705882]