import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns

Projection with dot product¶

${p} = {x} \frac{\langle {x}, {y} \rangle}{||{x}||^{2}}$

# We are trying to solve for the p-vector
u = np.array([[0,0],[2,4]])
v = np.array([[0,0],[5,5]])

def project(a,b):
    return a*(np.dot(a.T,b))/(np.dot(a.T,a))

p = project(v,u)
p

array([[0., 0.],
       [2., 4.]])

# Projection Matrix
P = np.outer(a,a.T)/np.dot(a.T, a)
P

array([[0.02173913, 0.04347826, 0.08695652, 0.10869565],
       [0.04347826, 0.08695652, 0.17391304, 0.2173913 ],
       [0.08695652, 0.17391304, 0.34782609, 0.43478261],
       [0.10869565, 0.2173913 , 0.43478261, 0.54347826]])

u = u.flatten()
v = v.flatten()
p = [2,4,1,-1] # This is our projection
plt.quiver([u[0], v[0], p[0]], 
           [u[1], v[1], p[1]], 
           [u[2], v[2], p[2]], 
           [u[3], v[3], p[3]],
           angles='xy', scale_units='xy', scale=1, color=sns.color_palette())
# plt.rc('text', usetex=True)
plt.xlim(-2, 6)
plt.ylim(-2, 8)
plt.axvline(x=0, color='grey')
plt.axhline(y=0, color='grey')

plt.text(0, 3, r'$||\vec{u}||$', color=sns.color_palette()[0], size=20)
plt.text(3, 1.5, r'$||\vec{v}||$', color=sns.color_palette()[1], size=20)
plt.text(2, 5, r'$||\vec{p}||$', color=sns.color_palette()[2], size=20)

plt.show()
plt.close()